Linear Regression Final Project

df <- as_tibble(read.table("winequality-red.csv", sep=";", header = TRUE))
df_white <- as_tibble(read.table("winequality-white.csv", sep=";", header = TRUE))

index = sample(1:nrow(df), 0.7*nrow(df)) 

train = df[index,] # Create the training data 
test = df[-index,] # Create the test data

dim(train)

## [1] 1119   12

dim(test)

## [1] 480  12

Rudimentary LM on whole dataset

lm_basic <- lm(data = df, quality ~ .)

summary(lm_basic)

## 
## Call:
## lm(formula = quality ~ ., data = df)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -2.68911 -0.36652 -0.04699  0.45202  2.02498 
## 
## Coefficients:
##                        Estimate Std. Error t value Pr(>|t|)    
## (Intercept)           2.197e+01  2.119e+01   1.036   0.3002    
## fixed.acidity         2.499e-02  2.595e-02   0.963   0.3357    
## volatile.acidity     -1.084e+00  1.211e-01  -8.948  < 2e-16 ***
## citric.acid          -1.826e-01  1.472e-01  -1.240   0.2150    
## residual.sugar        1.633e-02  1.500e-02   1.089   0.2765    
## chlorides            -1.874e+00  4.193e-01  -4.470 8.37e-06 ***
## free.sulfur.dioxide   4.361e-03  2.171e-03   2.009   0.0447 *  
## total.sulfur.dioxide -3.265e-03  7.287e-04  -4.480 8.00e-06 ***
## density              -1.788e+01  2.163e+01  -0.827   0.4086    
## pH                   -4.137e-01  1.916e-01  -2.159   0.0310 *  
## sulphates             9.163e-01  1.143e-01   8.014 2.13e-15 ***
## alcohol               2.762e-01  2.648e-02  10.429  < 2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.648 on 1587 degrees of freedom
## Multiple R-squared:  0.3606, Adjusted R-squared:  0.3561 
## F-statistic: 81.35 on 11 and 1587 DF,  p-value: < 2.2e-16

Volatile acidity, chlorides, total sulfur dioxide, sulphates and alcohol are all very statistically significant, although our adjusted \(R^2\) is still quite low.

Ridge regression

train_control <- trainControl(method  = "cv", number = 5)

model_ridge <- train(quality ~ .,
                     data = df,
                     method = "ridge",           # method
                     trControl = train_control)        # cross validation

model_ridge

## Ridge Regression 
## 
## 1599 samples
##   11 predictor
## 
## No pre-processing
## Resampling: Cross-Validated (5 fold) 
## Summary of sample sizes: 1280, 1280, 1279, 1279, 1278 
## Resampling results across tuning parameters:
## 
##   lambda  RMSE       Rsquared   MAE      
##   0e+00   0.6522813  0.3490324  0.5055778
##   1e-04   0.6522800  0.3490349  0.5055761
##   1e-01   0.6522864  0.3494269  0.5046582
## 
## RMSE was used to select the optimal model using the smallest value.
## The final value used for the model was lambda = 1e-04.

#something doesn't seem to be working here

model_stepwise <- train(quality ~ .,
                        data = df,
                        method = "glmStepAIC",
                        trControl = train_control)

## Start:  AIC=2545.06
## .outcome ~ fixed.acidity + volatile.acidity + citric.acid + residual.sugar + 
##     chlorides + free.sulfur.dioxide + total.sulfur.dioxide + 
##     density + pH + sulphates + alcohol
## 
##                        Df Deviance    AIC
## - density               1   537.24 2543.3
## - fixed.acidity         1   537.45 2543.8
## - residual.sugar        1   537.64 2544.2
## <none>                      537.15 2545.1
## - free.sulfur.dioxide   1   538.42 2546.1
## - citric.acid           1   538.73 2546.8
## - pH                    1   539.50 2548.6
## - chlorides             1   544.44 2560.3
## - total.sulfur.dioxide  1   544.72 2560.9
## - sulphates             1   560.75 2598.0
## - volatile.acidity      1   568.08 2614.6
## - alcohol               1   575.99 2632.3
## 
## Step:  AIC=2543.27
## .outcome ~ fixed.acidity + volatile.acidity + citric.acid + residual.sugar + 
##     chlorides + free.sulfur.dioxide + total.sulfur.dioxide + 
##     pH + sulphates + alcohol
## 
##                        Df Deviance    AIC
## - fixed.acidity         1   537.49 2541.9
## - residual.sugar        1   537.66 2542.3
## <none>                      537.24 2543.3
## - free.sulfur.dioxide   1   538.57 2544.4
## - citric.acid           1   538.84 2545.1
## - pH                    1   541.51 2551.4
## - chlorides             1   544.73 2559.0
## - total.sulfur.dioxide  1   544.99 2559.6
## - sulphates             1   561.62 2598.0
## - volatile.acidity      1   569.07 2614.8
## - alcohol               1   635.19 2755.3
## 
## Step:  AIC=2541.87
## .outcome ~ volatile.acidity + citric.acid + residual.sugar + 
##     chlorides + free.sulfur.dioxide + total.sulfur.dioxide + 
##     pH + sulphates + alcohol
## 
##                        Df Deviance    AIC
## - residual.sugar        1   537.96 2541.0
## <none>                      537.49 2541.9
## - citric.acid           1   538.90 2543.2
## - free.sulfur.dioxide   1   538.94 2543.3
## - pH                    1   544.82 2557.2
## - chlorides             1   546.51 2561.1
## - total.sulfur.dioxide  1   546.75 2561.7
## - sulphates             1   562.40 2597.8
## - volatile.acidity      1   570.03 2615.0
## - alcohol               1   635.77 2754.5
## 
## Step:  AIC=2540.99
## .outcome ~ volatile.acidity + citric.acid + chlorides + free.sulfur.dioxide + 
##     total.sulfur.dioxide + pH + sulphates + alcohol
## 
##                        Df Deviance    AIC
## <none>                      537.96 2541.0
## - citric.acid           1   539.16 2541.8
## - free.sulfur.dioxide   1   539.68 2543.1
## - pH                    1   545.36 2556.5
## - chlorides             1   546.82 2559.9
## - total.sulfur.dioxide  1   547.01 2560.3
## - sulphates             1   562.50 2596.0
## - volatile.acidity      1   570.04 2613.0
## - alcohol               1   638.05 2757.1
## Start:  AIC=2556.61
## .outcome ~ fixed.acidity + volatile.acidity + citric.acid + residual.sugar + 
##     chlorides + free.sulfur.dioxide + total.sulfur.dioxide + 
##     density + pH + sulphates + alcohol
## 
##                        Df Deviance    AIC
## - citric.acid           1   541.62 2554.6
## - fixed.acidity         1   541.65 2554.7
## - residual.sugar        1   541.70 2554.8
## - density               1   541.73 2554.9
## <none>                      541.61 2556.6
## - free.sulfur.dioxide   1   543.33 2558.7
## - pH                    1   544.00 2560.2
## - chlorides             1   547.55 2568.6
## - total.sulfur.dioxide  1   548.78 2571.4
## - sulphates             1   563.77 2605.9
## - volatile.acidity      1   566.03 2611.0
## - alcohol               1   576.37 2634.2
## 
## Step:  AIC=2554.63
## .outcome ~ fixed.acidity + volatile.acidity + residual.sugar + 
##     chlorides + free.sulfur.dioxide + total.sulfur.dioxide + 
##     density + pH + sulphates + alcohol
## 
##                        Df Deviance    AIC
## - fixed.acidity         1   541.68 2552.8
## - residual.sugar        1   541.71 2552.9
## - density               1   541.74 2552.9
## <none>                      541.62 2554.6
## - free.sulfur.dioxide   1   543.34 2556.7
## - pH                    1   544.03 2558.3
## - chlorides             1   547.86 2567.3
## - total.sulfur.dioxide  1   549.14 2570.3
## - sulphates             1   563.80 2604.0
## - volatile.acidity      1   575.66 2630.6
## - alcohol               1   577.73 2635.2
## 
## Step:  AIC=2552.77
## .outcome ~ volatile.acidity + residual.sugar + chlorides + free.sulfur.dioxide + 
##     total.sulfur.dioxide + density + pH + sulphates + alcohol
## 
##                        Df Deviance    AIC
## - residual.sugar        1   541.73 2550.9
## - density               1   541.74 2550.9
## <none>                      541.68 2552.8
## - free.sulfur.dioxide   1   543.46 2555.0
## - chlorides             1   548.21 2566.1
## - pH                    1   549.20 2568.4
## - total.sulfur.dioxide  1   549.65 2569.5
## - sulphates             1   564.09 2602.6
## - volatile.acidity      1   577.22 2632.1
## - alcohol               1   604.74 2691.6
## 
## Step:  AIC=2550.89
## .outcome ~ volatile.acidity + chlorides + free.sulfur.dioxide + 
##     total.sulfur.dioxide + density + pH + sulphates + alcohol
## 
##                        Df Deviance    AIC
## - density               1   541.75 2549.0
## <none>                      541.73 2550.9
## - free.sulfur.dioxide   1   543.60 2553.3
## - chlorides             1   548.21 2564.1
## - pH                    1   549.23 2566.5
## - total.sulfur.dioxide  1   549.69 2567.6
## - sulphates             1   564.36 2601.2
## - volatile.acidity      1   577.25 2630.1
## - alcohol               1   615.84 2712.9
## 
## Step:  AIC=2548.96
## .outcome ~ volatile.acidity + chlorides + free.sulfur.dioxide + 
##     total.sulfur.dioxide + pH + sulphates + alcohol
## 
##                        Df Deviance    AIC
## <none>                      541.75 2549.0
## - free.sulfur.dioxide   1   543.63 2551.4
## - chlorides             1   548.24 2562.2
## - pH                    1   549.52 2565.2
## - total.sulfur.dioxide  1   549.69 2565.6
## - sulphates             1   564.84 2600.3
## - volatile.acidity      1   577.34 2628.3
## - alcohol               1   640.10 2760.3
## Start:  AIC=2543.52
## .outcome ~ fixed.acidity + volatile.acidity + citric.acid + residual.sugar + 
##     chlorides + free.sulfur.dioxide + total.sulfur.dioxide + 
##     density + pH + sulphates + alcohol
## 
##                        Df Deviance    AIC
## - density               1   536.50 2541.5
## - residual.sugar        1   536.50 2541.5
## - fixed.acidity         1   536.68 2541.9
## <none>                      536.50 2543.5
## - citric.acid           1   537.57 2544.1
## - pH                    1   538.00 2545.1
## - free.sulfur.dioxide   1   538.87 2547.2
## - chlorides             1   543.90 2559.0
## - total.sulfur.dioxide  1   544.95 2561.5
## - sulphates             1   555.38 2585.7
## - volatile.acidity      1   569.42 2617.6
## - alcohol               1   577.06 2634.7
## 
## Step:  AIC=2541.52
## .outcome ~ fixed.acidity + volatile.acidity + citric.acid + residual.sugar + 
##     chlorides + free.sulfur.dioxide + total.sulfur.dioxide + 
##     pH + sulphates + alcohol
## 
##                        Df Deviance    AIC
## - residual.sugar        1   536.51 2539.5
## - fixed.acidity         1   537.05 2540.8
## <none>                      536.50 2541.5
## - citric.acid           1   537.57 2542.1
## - pH                    1   538.66 2544.7
## - free.sulfur.dioxide   1   538.88 2545.2
## - chlorides             1   543.97 2557.2
## - total.sulfur.dioxide  1   545.02 2559.7
## - sulphates             1   556.72 2586.8
## - volatile.acidity      1   569.77 2616.4
## - alcohol               1   629.32 2743.4
## 
## Step:  AIC=2539.54
## .outcome ~ fixed.acidity + volatile.acidity + citric.acid + chlorides + 
##     free.sulfur.dioxide + total.sulfur.dioxide + pH + sulphates + 
##     alcohol
## 
##                        Df Deviance    AIC
## - fixed.acidity         1   537.07 2538.9
## <none>                      536.51 2539.5
## - citric.acid           1   537.57 2540.1
## - pH                    1   538.67 2542.7
## - free.sulfur.dioxide   1   538.95 2543.3
## - chlorides             1   543.98 2555.2
## - total.sulfur.dioxide  1   545.04 2557.7
## - sulphates             1   556.76 2584.9
## - volatile.acidity      1   569.84 2614.6
## - alcohol               1   630.30 2743.4
## 
## Step:  AIC=2538.88
## .outcome ~ volatile.acidity + citric.acid + chlorides + free.sulfur.dioxide + 
##     total.sulfur.dioxide + pH + sulphates + alcohol
## 
##                        Df Deviance    AIC
## - citric.acid           1   537.61 2538.2
## <none>                      537.07 2538.9
## - free.sulfur.dioxide   1   539.67 2543.1
## - pH                    1   542.11 2548.8
## - chlorides             1   546.01 2558.0
## - total.sulfur.dioxide  1   547.35 2561.1
## - sulphates             1   557.68 2585.0
## - volatile.acidity      1   570.45 2613.9
## - alcohol               1   630.35 2741.6
## 
## Step:  AIC=2538.17
## .outcome ~ volatile.acidity + chlorides + free.sulfur.dioxide + 
##     total.sulfur.dioxide + pH + sulphates + alcohol
## 
##                        Df Deviance    AIC
## <none>                      537.61 2538.2
## - free.sulfur.dioxide   1   540.62 2543.3
## - pH                    1   542.22 2547.1
## - chlorides             1   547.29 2559.0
## - total.sulfur.dioxide  1   548.94 2562.8
## - sulphates             1   557.80 2583.3
## - volatile.acidity      1   575.73 2623.7
## - alcohol               1   631.71 2742.3
## Start:  AIC=2542.23
## .outcome ~ fixed.acidity + volatile.acidity + citric.acid + residual.sugar + 
##     chlorides + free.sulfur.dioxide + total.sulfur.dioxide + 
##     density + pH + sulphates + alcohol
## 
##                        Df Deviance    AIC
## - citric.acid           1   534.93 2540.7
## - pH                    1   535.31 2541.6
## - residual.sugar        1   535.49 2542.0
## <none>                      534.75 2542.2
## - density               1   535.60 2542.3
## - fixed.acidity         1   535.61 2542.3
## - free.sulfur.dioxide   1   536.09 2543.5
## - total.sulfur.dioxide  1   540.99 2555.1
## - chlorides             1   541.04 2555.2
## - volatile.acidity      1   554.52 2586.8
## - sulphates             1   554.72 2587.2
## - alcohol               1   566.09 2613.2
## 
## Step:  AIC=2540.68
## .outcome ~ fixed.acidity + volatile.acidity + residual.sugar + 
##     chlorides + free.sulfur.dioxide + total.sulfur.dioxide + 
##     density + pH + sulphates + alcohol
## 
##                        Df Deviance    AIC
## - pH                    1   535.46 2539.9
## - residual.sugar        1   535.63 2540.3
## - fixed.acidity         1   535.63 2540.4
## <none>                      534.93 2540.7
## - density               1   535.80 2540.8
## - free.sulfur.dioxide   1   536.48 2542.4
## - total.sulfur.dioxide  1   542.31 2556.2
## - chlorides             1   542.45 2556.6
## - sulphates             1   554.88 2585.6
## - volatile.acidity      1   559.93 2597.2
## - alcohol               1   566.22 2611.5
## 
## Step:  AIC=2539.94
## .outcome ~ fixed.acidity + volatile.acidity + residual.sugar + 
##     chlorides + free.sulfur.dioxide + total.sulfur.dioxide + 
##     density + sulphates + alcohol
## 
##                        Df Deviance    AIC
## <none>                      535.46 2539.9
## - residual.sugar        1   536.75 2541.0
## - free.sulfur.dioxide   1   536.81 2541.2
## - density               1   538.20 2544.5
## - fixed.acidity         1   540.40 2549.7
## - total.sulfur.dioxide  1   542.35 2554.3
## - chlorides             1   542.45 2554.6
## - sulphates             1   556.73 2587.8
## - volatile.acidity      1   560.68 2596.9
## - alcohol               1   572.70 2624.1
## Start:  AIC=2480.66
## .outcome ~ fixed.acidity + volatile.acidity + citric.acid + residual.sugar + 
##     chlorides + free.sulfur.dioxide + total.sulfur.dioxide + 
##     density + pH + sulphates + alcohol
## 
##                        Df Deviance    AIC
## - free.sulfur.dioxide   1   510.51 2479.9
## - fixed.acidity         1   510.54 2480.0
## <none>                      510.02 2480.7
## - density               1   510.85 2480.8
## - citric.acid           1   511.01 2481.2
## - pH                    1   511.33 2482.0
## - residual.sugar        1   511.66 2482.8
## - total.sulfur.dioxide  1   514.79 2490.6
## - chlorides             1   516.37 2494.5
## - sulphates             1   533.92 2537.3
## - volatile.acidity      1   537.34 2545.5
## - alcohol               1   546.67 2567.5
## 
## Step:  AIC=2479.9
## .outcome ~ fixed.acidity + volatile.acidity + citric.acid + residual.sugar + 
##     chlorides + total.sulfur.dioxide + density + pH + sulphates + 
##     alcohol
## 
##                        Df Deviance    AIC
## - fixed.acidity         1   511.17 2479.5
## <none>                      510.51 2479.9
## - density               1   511.48 2480.3
## - pH                    1   511.63 2480.7
## - citric.acid           1   511.81 2481.2
## - residual.sugar        1   512.48 2482.8
## - total.sulfur.dioxide  1   515.81 2491.1
## - chlorides             1   516.65 2493.2
## - sulphates             1   534.95 2537.8
## - volatile.acidity      1   540.02 2549.8
## - alcohol               1   546.91 2566.1
## 
## Step:  AIC=2479.54
## .outcome ~ volatile.acidity + citric.acid + residual.sugar + 
##     chlorides + total.sulfur.dioxide + density + pH + sulphates + 
##     alcohol
## 
##                        Df Deviance    AIC
## - density               1   511.48 2478.3
## <none>                      511.17 2479.5
## - citric.acid           1   512.02 2479.7
## - residual.sugar        1   512.54 2481.0
## - pH                    1   516.79 2491.5
## - total.sulfur.dioxide  1   517.55 2493.4
## - chlorides             1   518.70 2496.3
## - sulphates             1   534.99 2535.8
## - volatile.acidity      1   540.33 2548.6
## - alcohol               1   570.89 2619.0
## 
## Step:  AIC=2478.34
## .outcome ~ volatile.acidity + citric.acid + residual.sugar + 
##     chlorides + total.sulfur.dioxide + pH + sulphates + alcohol
## 
##                        Df Deviance    AIC
## <none>                      511.48 2478.3
## - residual.sugar        1   512.54 2479.0
## - citric.acid           1   513.21 2480.7
## - pH                    1   517.11 2490.3
## - total.sulfur.dioxide  1   517.55 2491.4
## - chlorides             1   518.72 2494.3
## - sulphates             1   535.26 2534.5
## - volatile.acidity      1   544.56 2556.6
## - alcohol               1   618.45 2719.4
## Start:  AIC=3164.28
## .outcome ~ fixed.acidity + volatile.acidity + citric.acid + residual.sugar + 
##     chlorides + free.sulfur.dioxide + total.sulfur.dioxide + 
##     density + pH + sulphates + alcohol
## 
##                        Df Deviance    AIC
## - density               1   666.70 3163.0
## - fixed.acidity         1   666.80 3163.2
## - residual.sugar        1   666.91 3163.5
## - citric.acid           1   667.06 3163.8
## <none>                      666.41 3164.3
## - free.sulfur.dioxide   1   668.10 3166.3
## - pH                    1   668.37 3167.0
## - chlorides             1   674.80 3182.3
## - total.sulfur.dioxide  1   674.84 3182.4
## - sulphates             1   693.38 3225.7
## - volatile.acidity      1   700.03 3241.0
## - alcohol               1   712.08 3268.3
## 
## Step:  AIC=3162.96
## .outcome ~ fixed.acidity + volatile.acidity + citric.acid + residual.sugar + 
##     chlorides + free.sulfur.dioxide + total.sulfur.dioxide + 
##     pH + sulphates + alcohol
## 
##                        Df Deviance    AIC
## - fixed.acidity         1   666.81 3161.2
## - residual.sugar        1   666.93 3161.5
## - citric.acid           1   667.35 3162.5
## <none>                      666.70 3163.0
## - free.sulfur.dioxide   1   668.53 3165.3
## - pH                    1   671.02 3171.3
## - total.sulfur.dioxide  1   675.43 3181.8
## - chlorides             1   675.46 3181.8
## - sulphates             1   693.98 3225.1
## - volatile.acidity      1   701.70 3242.8
## - alcohol               1   786.37 3424.9
## 
## Step:  AIC=3161.22
## .outcome ~ volatile.acidity + citric.acid + residual.sugar + 
##     chlorides + free.sulfur.dioxide + total.sulfur.dioxide + 
##     pH + sulphates + alcohol
## 
##                        Df Deviance    AIC
## - residual.sugar        1   667.06 3159.8
## - citric.acid           1   667.37 3160.6
## <none>                      666.81 3161.2
## - free.sulfur.dioxide   1   668.71 3163.8
## - pH                    1   673.87 3176.1
## - chlorides             1   676.75 3182.9
## - total.sulfur.dioxide  1   676.84 3183.1
## - sulphates             1   694.48 3224.2
## - volatile.acidity      1   703.04 3243.8
## - alcohol               1   787.44 3425.1
## 
## Step:  AIC=3159.84
## .outcome ~ volatile.acidity + citric.acid + chlorides + free.sulfur.dioxide + 
##     total.sulfur.dioxide + pH + sulphates + alcohol
## 
##                        Df Deviance    AIC
## - citric.acid           1   667.54 3159.0
## <none>                      667.06 3159.8
## - free.sulfur.dioxide   1   669.13 3162.8
## - pH                    1   674.20 3174.9
## - total.sulfur.dioxide  1   676.89 3181.2
## - chlorides             1   676.89 3181.2
## - sulphates             1   694.51 3222.3
## - volatile.acidity      1   703.04 3241.8
## - alcohol               1   789.73 3427.8
## 
## Step:  AIC=3158.98
## .outcome ~ volatile.acidity + chlorides + free.sulfur.dioxide + 
##     total.sulfur.dioxide + pH + sulphates + alcohol
## 
##                        Df Deviance    AIC
## <none>                      667.54 3159.0
## - free.sulfur.dioxide   1   669.93 3162.7
## - pH                    1   674.61 3173.8
## - total.sulfur.dioxide  1   678.32 3182.6
## - chlorides             1   678.35 3182.7
## - sulphates             1   694.60 3220.5
## - volatile.acidity      1   709.85 3255.3
## - alcohol               1   792.02 3430.4

#model_stepwise

Explore anything highly correlated:

correlation_df <- cor(df)

#correlation_df

# fixed acidity to ph and sulphates

correlation_df_melt <- melt(correlation_df)

gz <- ggplot(correlation_df_melt, mapping = aes(x = Var1, y = Var2, fill = value)) + 
  geom_tile() + 
  theme(axis.text.x = element_text(angle = 90, hjust = 1))+
  theme(text = element_text(size = 8)) + 
  ggtitle("Heat map for correlation") + 
  ylab("")+
  xlab("")+
  scale_fill_distiller(palette = "RdPu")

ggplotly(gz, tooltip = "text")

minimized_lm_df <- df %>% 
  select(-free.sulfur.dioxide, -citric.acid, -density, -volatile.acidity, -pH)

minimized_lm <- lm(data = minimized_lm_df, quality ~ .)

summary(minimized_lm)

## 
## Call:
## lm(formula = quality ~ ., data = minimized_lm_df)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -2.71086 -0.37485 -0.06023  0.48038  2.08242 
## 
## Coefficients:
##                        Estimate Std. Error t value Pr(>|t|)    
## (Intercept)           1.5204529  0.2113680   7.193 9.69e-13 ***
## fixed.acidity         0.0485196  0.0101132   4.798 1.76e-06 ***
## residual.sugar        0.0057865  0.0124562   0.465    0.642    
## chlorides            -2.3443435  0.4030377  -5.817 7.24e-09 ***
## total.sulfur.dioxide -0.0023418  0.0005467  -4.283 1.95e-05 ***
## sulphates             1.1850119  0.1116567  10.613  < 2e-16 ***
## alcohol               0.3100041  0.0171438  18.083  < 2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.6742 on 1592 degrees of freedom
## Multiple R-squared:  0.3057, Adjusted R-squared:  0.3031 
## F-statistic: 116.8 on 6 and 1592 DF,  p-value: < 2.2e-16

correlations

findCorrelation(
  cor(df),
  cutoff = 0.5,
  verbose = TRUE,
  names = TRUE
)

## Compare row 3  and column  1 with corr  0.672 
##   Means:  0.293 vs 0.19 so flagging column 3 
## Compare row 1  and column  8 with corr  0.668 
##   Means:  0.245 vs 0.172 so flagging column 1 
## Compare row 7  and column  6 with corr  0.668 
##   Means:  0.174 vs 0.169 so flagging column 7 
## All correlations <= 0.5

## [1] "citric.acid"          "fixed.acidity"        "total.sulfur.dioxide"

ggcorrplot(correlation_df, hc.order = TRUE, type = "lower")

df_less_acid <- df %>% 
  select(-citric.acid, -fixed.acidity)

lm_less_acid <- lm(data = df_less_acid, quality ~ .)
summary(lm_less_acid)

## 
## Call:
## lm(formula = quality ~ ., data = df_less_acid)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -2.66740 -0.37223 -0.04556  0.46264  2.03092 
## 
## Coefficients:
##                        Estimate Std. Error t value Pr(>|t|)    
## (Intercept)          12.0575013 12.0089257   1.004   0.3155    
## volatile.acidity     -1.0127925  0.1009279 -10.035  < 2e-16 ***
## residual.sugar        0.0115046  0.0134606   0.855   0.3929    
## chlorides            -2.0490938  0.3992190  -5.133 3.21e-07 ***
## free.sulfur.dioxide   0.0048741  0.0021385   2.279   0.0228 *  
## total.sulfur.dioxide -0.0035687  0.0006939  -5.143 3.03e-07 ***
## density              -7.5668358 11.8648390  -0.638   0.5237    
## pH                   -0.4919908  0.1210931  -4.063 5.08e-05 ***
## sulphates             0.9024434  0.1129657   7.989 2.60e-15 ***
## alcohol               0.2810022  0.0201992  13.912  < 2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.648 on 1589 degrees of freedom
## Multiple R-squared:  0.3598, Adjusted R-squared:  0.3562 
## F-statistic: 99.22 on 9 and 1589 DF,  p-value: < 2.2e-16

Lasso

plot(df)

Full scaled analysis

colnames(df)

##  [1] "fixed.acidity"        "volatile.acidity"     "citric.acid"         
##  [4] "residual.sugar"       "chlorides"            "free.sulfur.dioxide" 
##  [7] "total.sulfur.dioxide" "density"              "pH"                  
## [10] "sulphates"            "alcohol"              "quality"

cols = c('fixed.acidity', 'volatile.acidity', 'citric.acid', 'residual.sugar', 'chlorides', 'free.sulfur.dioxide', 'total.sulfur.dioxide', 'density', 'pH', 'sulphates', 'alcohol')

pre_proc_val <- preProcess(train[,cols], method = c("center", "scale"))

train[,cols] = predict(pre_proc_val, train[,cols])
test[,cols] = predict(pre_proc_val, test[,cols])

summary(train)

##  fixed.acidity     volatile.acidity    citric.acid       residual.sugar    
##  Min.   :-1.8982   Min.   :-2.26862   Min.   :-1.40589   Min.   :-1.16322  
##  1st Qu.:-0.7070   1st Qu.:-0.77506   1st Qu.:-0.88988   1st Qu.:-0.45567  
##  Median :-0.2532   Median :-0.05593   Median :-0.06428   Median :-0.24341  
##  Mean   : 0.0000   Mean   : 0.00000   Mean   : 0.00000   Mean   : 0.00000  
##  3rd Qu.: 0.5410   3rd Qu.: 0.60787   3rd Qu.: 0.76132   3rd Qu.: 0.03961  
##  Max.   : 4.2850   Max.   : 5.80767   Max.   : 3.75413   Max.   : 9.16702  
##    chlorides        free.sulfur.dioxide total.sulfur.dioxide    density        
##  Min.   :-1.52980   Min.   :-1.4505     Min.   :-1.2541      Min.   :-3.52133  
##  1st Qu.:-0.36645   1st Qu.:-0.8664     1st Qu.:-0.7603      1st Qu.:-0.60428  
##  Median :-0.16587   Median :-0.1849     Median :-0.2665      Median : 0.01593  
##  Mean   : 0.00000   Mean   : 0.0000     Mean   : 0.0000      Mean   : 0.00000  
##  3rd Qu.: 0.05476   3rd Qu.: 0.4966     3rd Qu.: 0.4742      3rd Qu.: 0.59408  
##  Max.   :10.48479   Max.   : 5.0723     Max.   : 7.4799      Max.   : 3.63728  
##        pH             sulphates          alcohol           quality    
##  Min.   :-3.65381   Min.   :-1.8950   Min.   :-1.9054   Min.   :3.00  
##  1st Qu.:-0.63558   1st Qu.:-0.6400   1st Qu.:-0.8654   1st Qu.:5.00  
##  Median :-0.05762   Median :-0.2406   Median :-0.2036   Median :6.00  
##  Mean   : 0.00000   Mean   : 0.0000   Mean   : 0.0000   Mean   :5.63  
##  3rd Qu.: 0.58456   3rd Qu.: 0.3869   3rd Qu.: 0.6473   3rd Qu.:6.00  
##  Max.   : 4.50184   Max.   : 7.6320   Max.   : 4.2400   Max.   :8.00

#generic LM

lm_2 <- lm(quality ~ ., data = train)
summary(lm_2)

## 
## Call:
## lm(formula = quality ~ ., data = train)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -2.67679 -0.36806 -0.05664  0.44772  1.98812 
## 
## Coefficients:
##                       Estimate Std. Error t value Pr(>|t|)    
## (Intercept)           5.630027   0.019547 288.030  < 2e-16 ***
## fixed.acidity         0.005020   0.053098   0.095    0.925    
## volatile.acidity     -0.182301   0.025788  -7.069 2.76e-12 ***
## citric.acid           0.006733   0.033395   0.202    0.840    
## residual.sugar        0.022515   0.024818   0.907    0.365    
## chlorides            -0.103942   0.023933  -4.343 1.53e-05 ***
## free.sulfur.dioxide   0.037472   0.027207   1.377    0.169    
## total.sulfur.dioxide -0.117017   0.028773  -4.067 5.10e-05 ***
## density              -0.016108   0.048469  -0.332    0.740    
## pH                   -0.038438   0.035558  -1.081    0.280    
## sulphates             0.161686   0.023577   6.858 1.16e-11 ***
## alcohol               0.273154   0.033538   8.145 1.02e-15 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.6539 on 1107 degrees of freedom
## Multiple R-squared:  0.3452, Adjusted R-squared:  0.3387 
## F-statistic: 53.06 on 11 and 1107 DF,  p-value: < 2.2e-16

Regularization

cols_reg = c('fixed.acidity', 'volatile.acidity', 'citric.acid', 'residual.sugar', 'chlorides', 'free.sulfur.dioxide', 'total.sulfur.dioxide', 'density', 'pH', 'sulphates', 'alcohol', 'quality')

dummies <- dummyVars(quality ~ ., data = df[,cols_reg])

train_dummies = predict(dummies, newdata = train[,cols_reg])

test_dummies = predict(dummies, newdata = test[,cols_reg])

print(dim(train_dummies)); print(dim(test_dummies))

## [1] 1119   11

## [1] 480  11

Ridge Regression

x = as.matrix(train_dummies)
y_train = train$quality

x_test = as.matrix(test_dummies)
y_test = test$quality

lambdas <- 10^seq(2, -3, by = -.1)
ridge_reg = glmnet(x, y_train, nlambda = 25, alpha = 0, family = 'gaussian', lambda = lambdas)

summary(ridge_reg)

##           Length Class     Mode   
## a0         51    -none-    numeric
## beta      561    dgCMatrix S4     
## df         51    -none-    numeric
## dim         2    -none-    numeric
## lambda     51    -none-    numeric
## dev.ratio  51    -none-    numeric
## nulldev     1    -none-    numeric
## npasses     1    -none-    numeric
## jerr        1    -none-    numeric
## offset      1    -none-    logical
## call        7    -none-    call   
## nobs        1    -none-    numeric

Optimal Lambda

cv_ridge <- cv.glmnet(x, y_train, alpha = 0, lambda = lambdas)
optimal_lambda <- cv_ridge$lambda.min
optimal_lambda

## [1] 0.05011872

Formal metrics for Ridge

# Compute R^2 from true and predicted values
eval_results <- function(true, predicted, df) {
  SSE <- sum((predicted - true)^2)
  SST <- sum((true - mean(true))^2)
  R_square <- 1 - SSE / SST
  RMSE = sqrt(SSE/nrow(df))

  
  # Model performance metrics
data.frame(
  RMSE = RMSE,
  Rsquare = R_square
)
  
}

# Prediction and evaluation on train data
predictions_train <- predict(ridge_reg, s = optimal_lambda, newx = x)
eval_results(y_train, predictions_train, train)

##        RMSE   Rsquare
## 1 0.6508117 0.3443019

# Prediction and evaluation on test data
predictions_test <- predict(ridge_reg, s = optimal_lambda, newx = x_test)
eval_results(y_test, predictions_test, test)

##        RMSE   Rsquare
## 1 0.6415514 0.3810703

Lasso

lambdas <- 10^seq(2, -3, by = -.1)

# Setting alpha = 1 implements lasso regression
lasso_reg <- cv.glmnet(x, y_train, alpha = 1, lambda = lambdas, standardize = TRUE, nfolds = 5)

# Best 
lambda_best <- lasso_reg$lambda.min 
lambda_best

## [1] 0.01584893

lasso_model <- glmnet(x, y_train, alpha = 1, lambda = lambda_best, standardize = TRUE)

predictions_train <- predict(lasso_model, s = lambda_best, newx = x)
eval_results(y_train, predictions_train, train)

##       RMSE   Rsquare
## 1 0.652377 0.3411439

predictions_test <- predict(lasso_model, s = lambda_best, newx = x_test)
eval_results(y_test, predictions_test, test)

##        RMSE   Rsquare
## 1 0.6450133 0.3743728

Mix of ridge and lasso (elasticnet)

# Set training control
train_cont <- trainControl(method = "repeatedcv",
                              number = 10,
                              repeats = 5,
                              search = "random",
                              verboseIter = TRUE)

# Train the model
elastic_reg <- train(quality ~ .,
                           data = train,
                           method = "glmnet",
                           preProcess = c("center", "scale"),
                           tuneLength = 10,
                           trControl = train_cont)

## + Fold01.Rep1: alpha=0.64457, lambda=0.002748 
## - Fold01.Rep1: alpha=0.64457, lambda=0.002748 
## + Fold01.Rep1: alpha=0.56029, lambda=0.304452 
## - Fold01.Rep1: alpha=0.56029, lambda=0.304452 
## + Fold01.Rep1: alpha=0.34217, lambda=3.683040 
## - Fold01.Rep1: alpha=0.34217, lambda=3.683040 
## + Fold01.Rep1: alpha=0.76305, lambda=0.037659 
## - Fold01.Rep1: alpha=0.76305, lambda=0.037659 
## + Fold01.Rep1: alpha=0.32620, lambda=0.010929 
## - Fold01.Rep1: alpha=0.32620, lambda=0.010929 
## + Fold01.Rep1: alpha=0.44663, lambda=0.001037 
## - Fold01.Rep1: alpha=0.44663, lambda=0.001037 
## + Fold01.Rep1: alpha=0.08757, lambda=0.012689 
## - Fold01.Rep1: alpha=0.08757, lambda=0.012689 
## + Fold01.Rep1: alpha=0.60040, lambda=0.003594 
## - Fold01.Rep1: alpha=0.60040, lambda=0.003594 
## + Fold01.Rep1: alpha=0.27368, lambda=0.453264 
## - Fold01.Rep1: alpha=0.27368, lambda=0.453264 
## + Fold01.Rep1: alpha=0.36848, lambda=4.961816 
## - Fold01.Rep1: alpha=0.36848, lambda=4.961816 
## + Fold02.Rep1: alpha=0.64457, lambda=0.002748 
## - Fold02.Rep1: alpha=0.64457, lambda=0.002748 
## + Fold02.Rep1: alpha=0.56029, lambda=0.304452 
## - Fold02.Rep1: alpha=0.56029, lambda=0.304452 
## + Fold02.Rep1: alpha=0.34217, lambda=3.683040 
## - Fold02.Rep1: alpha=0.34217, lambda=3.683040 
## + Fold02.Rep1: alpha=0.76305, lambda=0.037659 
## - Fold02.Rep1: alpha=0.76305, lambda=0.037659 
## + Fold02.Rep1: alpha=0.32620, lambda=0.010929 
## - Fold02.Rep1: alpha=0.32620, lambda=0.010929 
## + Fold02.Rep1: alpha=0.44663, lambda=0.001037 
## - Fold02.Rep1: alpha=0.44663, lambda=0.001037 
## + Fold02.Rep1: alpha=0.08757, lambda=0.012689 
## - Fold02.Rep1: alpha=0.08757, lambda=0.012689 
## + Fold02.Rep1: alpha=0.60040, lambda=0.003594 
## - Fold02.Rep1: alpha=0.60040, lambda=0.003594 
## + Fold02.Rep1: alpha=0.27368, lambda=0.453264 
## - Fold02.Rep1: alpha=0.27368, lambda=0.453264 
## + Fold02.Rep1: alpha=0.36848, lambda=4.961816 
## - Fold02.Rep1: alpha=0.36848, lambda=4.961816 
## + Fold03.Rep1: alpha=0.64457, lambda=0.002748 
## - Fold03.Rep1: alpha=0.64457, lambda=0.002748 
## + Fold03.Rep1: alpha=0.56029, lambda=0.304452 
## - Fold03.Rep1: alpha=0.56029, lambda=0.304452 
## + Fold03.Rep1: alpha=0.34217, lambda=3.683040 
## - Fold03.Rep1: alpha=0.34217, lambda=3.683040 
## + Fold03.Rep1: alpha=0.76305, lambda=0.037659 
## - Fold03.Rep1: alpha=0.76305, lambda=0.037659 
## + Fold03.Rep1: alpha=0.32620, lambda=0.010929 
## - Fold03.Rep1: alpha=0.32620, lambda=0.010929 
## + Fold03.Rep1: alpha=0.44663, lambda=0.001037 
## - Fold03.Rep1: alpha=0.44663, lambda=0.001037 
## + Fold03.Rep1: alpha=0.08757, lambda=0.012689 
## - Fold03.Rep1: alpha=0.08757, lambda=0.012689 
## + Fold03.Rep1: alpha=0.60040, lambda=0.003594 
## - Fold03.Rep1: alpha=0.60040, lambda=0.003594 
## + Fold03.Rep1: alpha=0.27368, lambda=0.453264 
## - Fold03.Rep1: alpha=0.27368, lambda=0.453264 
## + Fold03.Rep1: alpha=0.36848, lambda=4.961816 
## - Fold03.Rep1: alpha=0.36848, lambda=4.961816 
## + Fold04.Rep1: alpha=0.64457, lambda=0.002748 
## - Fold04.Rep1: alpha=0.64457, lambda=0.002748 
## + Fold04.Rep1: alpha=0.56029, lambda=0.304452 
## - Fold04.Rep1: alpha=0.56029, lambda=0.304452 
## + Fold04.Rep1: alpha=0.34217, lambda=3.683040 
## - Fold04.Rep1: alpha=0.34217, lambda=3.683040 
## + Fold04.Rep1: alpha=0.76305, lambda=0.037659 
## - Fold04.Rep1: alpha=0.76305, lambda=0.037659 
## + Fold04.Rep1: alpha=0.32620, lambda=0.010929 
## - Fold04.Rep1: alpha=0.32620, lambda=0.010929 
## + Fold04.Rep1: alpha=0.44663, lambda=0.001037 
## - Fold04.Rep1: alpha=0.44663, lambda=0.001037 
## + Fold04.Rep1: alpha=0.08757, lambda=0.012689 
## - Fold04.Rep1: alpha=0.08757, lambda=0.012689 
## + Fold04.Rep1: alpha=0.60040, lambda=0.003594 
## - Fold04.Rep1: alpha=0.60040, lambda=0.003594 
## + Fold04.Rep1: alpha=0.27368, lambda=0.453264 
## - Fold04.Rep1: alpha=0.27368, lambda=0.453264 
## + Fold04.Rep1: alpha=0.36848, lambda=4.961816 
## - Fold04.Rep1: alpha=0.36848, lambda=4.961816 
## + Fold05.Rep1: alpha=0.64457, lambda=0.002748 
## - Fold05.Rep1: alpha=0.64457, lambda=0.002748 
## + Fold05.Rep1: alpha=0.56029, lambda=0.304452 
## - Fold05.Rep1: alpha=0.56029, lambda=0.304452 
## + Fold05.Rep1: alpha=0.34217, lambda=3.683040 
## - Fold05.Rep1: alpha=0.34217, lambda=3.683040 
## + Fold05.Rep1: alpha=0.76305, lambda=0.037659 
## - Fold05.Rep1: alpha=0.76305, lambda=0.037659 
## + Fold05.Rep1: alpha=0.32620, lambda=0.010929 
## - Fold05.Rep1: alpha=0.32620, lambda=0.010929 
## + Fold05.Rep1: alpha=0.44663, lambda=0.001037 
## - Fold05.Rep1: alpha=0.44663, lambda=0.001037 
## + Fold05.Rep1: alpha=0.08757, lambda=0.012689 
## - Fold05.Rep1: alpha=0.08757, lambda=0.012689 
## + Fold05.Rep1: alpha=0.60040, lambda=0.003594 
## - Fold05.Rep1: alpha=0.60040, lambda=0.003594 
## + Fold05.Rep1: alpha=0.27368, lambda=0.453264 
## - Fold05.Rep1: alpha=0.27368, lambda=0.453264 
## + Fold05.Rep1: alpha=0.36848, lambda=4.961816 
## - Fold05.Rep1: alpha=0.36848, lambda=4.961816 
## + Fold06.Rep1: alpha=0.64457, lambda=0.002748 
## - Fold06.Rep1: alpha=0.64457, lambda=0.002748 
## + Fold06.Rep1: alpha=0.56029, lambda=0.304452 
## - Fold06.Rep1: alpha=0.56029, lambda=0.304452 
## + Fold06.Rep1: alpha=0.34217, lambda=3.683040 
## - Fold06.Rep1: alpha=0.34217, lambda=3.683040 
## + Fold06.Rep1: alpha=0.76305, lambda=0.037659 
## - Fold06.Rep1: alpha=0.76305, lambda=0.037659 
## + Fold06.Rep1: alpha=0.32620, lambda=0.010929 
## - Fold06.Rep1: alpha=0.32620, lambda=0.010929 
## + Fold06.Rep1: alpha=0.44663, lambda=0.001037 
## - Fold06.Rep1: alpha=0.44663, lambda=0.001037 
## + Fold06.Rep1: alpha=0.08757, lambda=0.012689 
## - Fold06.Rep1: alpha=0.08757, lambda=0.012689 
## + Fold06.Rep1: alpha=0.60040, lambda=0.003594 
## - Fold06.Rep1: alpha=0.60040, lambda=0.003594 
## + Fold06.Rep1: alpha=0.27368, lambda=0.453264 
## - Fold06.Rep1: alpha=0.27368, lambda=0.453264 
## + Fold06.Rep1: alpha=0.36848, lambda=4.961816 
## - Fold06.Rep1: alpha=0.36848, lambda=4.961816 
## + Fold07.Rep1: alpha=0.64457, lambda=0.002748 
## - Fold07.Rep1: alpha=0.64457, lambda=0.002748 
## + Fold07.Rep1: alpha=0.56029, lambda=0.304452 
## - Fold07.Rep1: alpha=0.56029, lambda=0.304452 
## + Fold07.Rep1: alpha=0.34217, lambda=3.683040 
## - Fold07.Rep1: alpha=0.34217, lambda=3.683040 
## + Fold07.Rep1: alpha=0.76305, lambda=0.037659 
## - Fold07.Rep1: alpha=0.76305, lambda=0.037659 
## + Fold07.Rep1: alpha=0.32620, lambda=0.010929 
## - Fold07.Rep1: alpha=0.32620, lambda=0.010929 
## + Fold07.Rep1: alpha=0.44663, lambda=0.001037 
## - Fold07.Rep1: alpha=0.44663, lambda=0.001037 
## + Fold07.Rep1: alpha=0.08757, lambda=0.012689 
## - Fold07.Rep1: alpha=0.08757, lambda=0.012689 
## + Fold07.Rep1: alpha=0.60040, lambda=0.003594 
## - Fold07.Rep1: alpha=0.60040, lambda=0.003594 
## + Fold07.Rep1: alpha=0.27368, lambda=0.453264 
## - Fold07.Rep1: alpha=0.27368, lambda=0.453264 
## + Fold07.Rep1: alpha=0.36848, lambda=4.961816 
## - Fold07.Rep1: alpha=0.36848, lambda=4.961816 
## + Fold08.Rep1: alpha=0.64457, lambda=0.002748 
## - Fold08.Rep1: alpha=0.64457, lambda=0.002748 
## + Fold08.Rep1: alpha=0.56029, lambda=0.304452 
## - Fold08.Rep1: alpha=0.56029, lambda=0.304452 
## + Fold08.Rep1: alpha=0.34217, lambda=3.683040 
## - Fold08.Rep1: alpha=0.34217, lambda=3.683040 
## + Fold08.Rep1: alpha=0.76305, lambda=0.037659 
## - Fold08.Rep1: alpha=0.76305, lambda=0.037659 
## + Fold08.Rep1: alpha=0.32620, lambda=0.010929 
## - Fold08.Rep1: alpha=0.32620, lambda=0.010929 
## + Fold08.Rep1: alpha=0.44663, lambda=0.001037 
## - Fold08.Rep1: alpha=0.44663, lambda=0.001037 
## + Fold08.Rep1: alpha=0.08757, lambda=0.012689 
## - Fold08.Rep1: alpha=0.08757, lambda=0.012689 
## + Fold08.Rep1: alpha=0.60040, lambda=0.003594 
## - Fold08.Rep1: alpha=0.60040, lambda=0.003594 
## + Fold08.Rep1: alpha=0.27368, lambda=0.453264 
## - Fold08.Rep1: alpha=0.27368, lambda=0.453264 
## + Fold08.Rep1: alpha=0.36848, lambda=4.961816 
## - Fold08.Rep1: alpha=0.36848, lambda=4.961816 
## + Fold09.Rep1: alpha=0.64457, lambda=0.002748 
## - Fold09.Rep1: alpha=0.64457, lambda=0.002748 
## + Fold09.Rep1: alpha=0.56029, lambda=0.304452 
## - Fold09.Rep1: alpha=0.56029, lambda=0.304452 
## + Fold09.Rep1: alpha=0.34217, lambda=3.683040 
## - Fold09.Rep1: alpha=0.34217, lambda=3.683040 
## + Fold09.Rep1: alpha=0.76305, lambda=0.037659 
## - Fold09.Rep1: alpha=0.76305, lambda=0.037659 
## + Fold09.Rep1: alpha=0.32620, lambda=0.010929 
## - Fold09.Rep1: alpha=0.32620, lambda=0.010929 
## + Fold09.Rep1: alpha=0.44663, lambda=0.001037 
## - Fold09.Rep1: alpha=0.44663, lambda=0.001037 
## + Fold09.Rep1: alpha=0.08757, lambda=0.012689 
## - Fold09.Rep1: alpha=0.08757, lambda=0.012689 
## + Fold09.Rep1: alpha=0.60040, lambda=0.003594 
## - Fold09.Rep1: alpha=0.60040, lambda=0.003594 
## + Fold09.Rep1: alpha=0.27368, lambda=0.453264 
## - Fold09.Rep1: alpha=0.27368, lambda=0.453264 
## + Fold09.Rep1: alpha=0.36848, lambda=4.961816 
## - Fold09.Rep1: alpha=0.36848, lambda=4.961816 
## + Fold10.Rep1: alpha=0.64457, lambda=0.002748 
## - Fold10.Rep1: alpha=0.64457, lambda=0.002748 
## + Fold10.Rep1: alpha=0.56029, lambda=0.304452 
## - Fold10.Rep1: alpha=0.56029, lambda=0.304452 
## + Fold10.Rep1: alpha=0.34217, lambda=3.683040 
## - Fold10.Rep1: alpha=0.34217, lambda=3.683040 
## + Fold10.Rep1: alpha=0.76305, lambda=0.037659 
## - Fold10.Rep1: alpha=0.76305, lambda=0.037659 
## + Fold10.Rep1: alpha=0.32620, lambda=0.010929 
## - Fold10.Rep1: alpha=0.32620, lambda=0.010929 
## + Fold10.Rep1: alpha=0.44663, lambda=0.001037 
## - Fold10.Rep1: alpha=0.44663, lambda=0.001037 
## + Fold10.Rep1: alpha=0.08757, lambda=0.012689 
## - Fold10.Rep1: alpha=0.08757, lambda=0.012689 
## + Fold10.Rep1: alpha=0.60040, lambda=0.003594 
## - Fold10.Rep1: alpha=0.60040, lambda=0.003594 
## + Fold10.Rep1: alpha=0.27368, lambda=0.453264 
## - Fold10.Rep1: alpha=0.27368, lambda=0.453264 
## + Fold10.Rep1: alpha=0.36848, lambda=4.961816 
## - Fold10.Rep1: alpha=0.36848, lambda=4.961816 
## + Fold01.Rep2: alpha=0.64457, lambda=0.002748 
## - Fold01.Rep2: alpha=0.64457, lambda=0.002748 
## + Fold01.Rep2: alpha=0.56029, lambda=0.304452 
## - Fold01.Rep2: alpha=0.56029, lambda=0.304452 
## + Fold01.Rep2: alpha=0.34217, lambda=3.683040 
## - Fold01.Rep2: alpha=0.34217, lambda=3.683040 
## + Fold01.Rep2: alpha=0.76305, lambda=0.037659 
## - Fold01.Rep2: alpha=0.76305, lambda=0.037659 
## + Fold01.Rep2: alpha=0.32620, lambda=0.010929 
## - Fold01.Rep2: alpha=0.32620, lambda=0.010929 
## + Fold01.Rep2: alpha=0.44663, lambda=0.001037 
## - Fold01.Rep2: alpha=0.44663, lambda=0.001037 
## + Fold01.Rep2: alpha=0.08757, lambda=0.012689 
## - Fold01.Rep2: alpha=0.08757, lambda=0.012689 
## + Fold01.Rep2: alpha=0.60040, lambda=0.003594 
## - Fold01.Rep2: alpha=0.60040, lambda=0.003594 
## + Fold01.Rep2: alpha=0.27368, lambda=0.453264 
## - Fold01.Rep2: alpha=0.27368, lambda=0.453264 
## + Fold01.Rep2: alpha=0.36848, lambda=4.961816 
## - Fold01.Rep2: alpha=0.36848, lambda=4.961816 
## + Fold02.Rep2: alpha=0.64457, lambda=0.002748 
## - Fold02.Rep2: alpha=0.64457, lambda=0.002748 
## + Fold02.Rep2: alpha=0.56029, lambda=0.304452 
## - Fold02.Rep2: alpha=0.56029, lambda=0.304452 
## + Fold02.Rep2: alpha=0.34217, lambda=3.683040 
## - Fold02.Rep2: alpha=0.34217, lambda=3.683040 
## + Fold02.Rep2: alpha=0.76305, lambda=0.037659 
## - Fold02.Rep2: alpha=0.76305, lambda=0.037659 
## + Fold02.Rep2: alpha=0.32620, lambda=0.010929 
## - Fold02.Rep2: alpha=0.32620, lambda=0.010929 
## + Fold02.Rep2: alpha=0.44663, lambda=0.001037 
## - Fold02.Rep2: alpha=0.44663, lambda=0.001037 
## + Fold02.Rep2: alpha=0.08757, lambda=0.012689 
## - Fold02.Rep2: alpha=0.08757, lambda=0.012689 
## + Fold02.Rep2: alpha=0.60040, lambda=0.003594 
## - Fold02.Rep2: alpha=0.60040, lambda=0.003594 
## + Fold02.Rep2: alpha=0.27368, lambda=0.453264 
## - Fold02.Rep2: alpha=0.27368, lambda=0.453264 
## + Fold02.Rep2: alpha=0.36848, lambda=4.961816 
## - Fold02.Rep2: alpha=0.36848, lambda=4.961816 
## + Fold03.Rep2: alpha=0.64457, lambda=0.002748 
## - Fold03.Rep2: alpha=0.64457, lambda=0.002748 
## + Fold03.Rep2: alpha=0.56029, lambda=0.304452 
## - Fold03.Rep2: alpha=0.56029, lambda=0.304452 
## + Fold03.Rep2: alpha=0.34217, lambda=3.683040 
## - Fold03.Rep2: alpha=0.34217, lambda=3.683040 
## + Fold03.Rep2: alpha=0.76305, lambda=0.037659 
## - Fold03.Rep2: alpha=0.76305, lambda=0.037659 
## + Fold03.Rep2: alpha=0.32620, lambda=0.010929 
## - Fold03.Rep2: alpha=0.32620, lambda=0.010929 
## + Fold03.Rep2: alpha=0.44663, lambda=0.001037 
## - Fold03.Rep2: alpha=0.44663, lambda=0.001037 
## + Fold03.Rep2: alpha=0.08757, lambda=0.012689 
## - Fold03.Rep2: alpha=0.08757, lambda=0.012689 
## + Fold03.Rep2: alpha=0.60040, lambda=0.003594 
## - Fold03.Rep2: alpha=0.60040, lambda=0.003594 
## + Fold03.Rep2: alpha=0.27368, lambda=0.453264 
## - Fold03.Rep2: alpha=0.27368, lambda=0.453264 
## + Fold03.Rep2: alpha=0.36848, lambda=4.961816 
## - Fold03.Rep2: alpha=0.36848, lambda=4.961816 
## + Fold04.Rep2: alpha=0.64457, lambda=0.002748 
## - Fold04.Rep2: alpha=0.64457, lambda=0.002748 
## + Fold04.Rep2: alpha=0.56029, lambda=0.304452 
## - Fold04.Rep2: alpha=0.56029, lambda=0.304452 
## + Fold04.Rep2: alpha=0.34217, lambda=3.683040 
## - Fold04.Rep2: alpha=0.34217, lambda=3.683040 
## + Fold04.Rep2: alpha=0.76305, lambda=0.037659 
## - Fold04.Rep2: alpha=0.76305, lambda=0.037659 
## + Fold04.Rep2: alpha=0.32620, lambda=0.010929 
## - Fold04.Rep2: alpha=0.32620, lambda=0.010929 
## + Fold04.Rep2: alpha=0.44663, lambda=0.001037 
## - Fold04.Rep2: alpha=0.44663, lambda=0.001037 
## + Fold04.Rep2: alpha=0.08757, lambda=0.012689 
## - Fold04.Rep2: alpha=0.08757, lambda=0.012689 
## + Fold04.Rep2: alpha=0.60040, lambda=0.003594 
## - Fold04.Rep2: alpha=0.60040, lambda=0.003594 
## + Fold04.Rep2: alpha=0.27368, lambda=0.453264 
## - Fold04.Rep2: alpha=0.27368, lambda=0.453264 
## + Fold04.Rep2: alpha=0.36848, lambda=4.961816 
## - Fold04.Rep2: alpha=0.36848, lambda=4.961816 
## + Fold05.Rep2: alpha=0.64457, lambda=0.002748 
## - Fold05.Rep2: alpha=0.64457, lambda=0.002748 
## + Fold05.Rep2: alpha=0.56029, lambda=0.304452 
## - Fold05.Rep2: alpha=0.56029, lambda=0.304452 
## + Fold05.Rep2: alpha=0.34217, lambda=3.683040 
## - Fold05.Rep2: alpha=0.34217, lambda=3.683040 
## + Fold05.Rep2: alpha=0.76305, lambda=0.037659 
## - Fold05.Rep2: alpha=0.76305, lambda=0.037659 
## + Fold05.Rep2: alpha=0.32620, lambda=0.010929 
## - Fold05.Rep2: alpha=0.32620, lambda=0.010929 
## + Fold05.Rep2: alpha=0.44663, lambda=0.001037 
## - Fold05.Rep2: alpha=0.44663, lambda=0.001037 
## + Fold05.Rep2: alpha=0.08757, lambda=0.012689 
## - Fold05.Rep2: alpha=0.08757, lambda=0.012689 
## + Fold05.Rep2: alpha=0.60040, lambda=0.003594 
## - Fold05.Rep2: alpha=0.60040, lambda=0.003594 
## + Fold05.Rep2: alpha=0.27368, lambda=0.453264 
## - Fold05.Rep2: alpha=0.27368, lambda=0.453264 
## + Fold05.Rep2: alpha=0.36848, lambda=4.961816 
## - Fold05.Rep2: alpha=0.36848, lambda=4.961816 
## + Fold06.Rep2: alpha=0.64457, lambda=0.002748 
## - Fold06.Rep2: alpha=0.64457, lambda=0.002748 
## + Fold06.Rep2: alpha=0.56029, lambda=0.304452 
## - Fold06.Rep2: alpha=0.56029, lambda=0.304452 
## + Fold06.Rep2: alpha=0.34217, lambda=3.683040 
## - Fold06.Rep2: alpha=0.34217, lambda=3.683040 
## + Fold06.Rep2: alpha=0.76305, lambda=0.037659 
## - Fold06.Rep2: alpha=0.76305, lambda=0.037659 
## + Fold06.Rep2: alpha=0.32620, lambda=0.010929 
## - Fold06.Rep2: alpha=0.32620, lambda=0.010929 
## + Fold06.Rep2: alpha=0.44663, lambda=0.001037 
## - Fold06.Rep2: alpha=0.44663, lambda=0.001037 
## + Fold06.Rep2: alpha=0.08757, lambda=0.012689 
## - Fold06.Rep2: alpha=0.08757, lambda=0.012689 
## + Fold06.Rep2: alpha=0.60040, lambda=0.003594 
## - Fold06.Rep2: alpha=0.60040, lambda=0.003594 
## + Fold06.Rep2: alpha=0.27368, lambda=0.453264 
## - Fold06.Rep2: alpha=0.27368, lambda=0.453264 
## + Fold06.Rep2: alpha=0.36848, lambda=4.961816 
## - Fold06.Rep2: alpha=0.36848, lambda=4.961816 
## + Fold07.Rep2: alpha=0.64457, lambda=0.002748 
## - Fold07.Rep2: alpha=0.64457, lambda=0.002748 
## + Fold07.Rep2: alpha=0.56029, lambda=0.304452 
## - Fold07.Rep2: alpha=0.56029, lambda=0.304452 
## + Fold07.Rep2: alpha=0.34217, lambda=3.683040 
## - Fold07.Rep2: alpha=0.34217, lambda=3.683040 
## + Fold07.Rep2: alpha=0.76305, lambda=0.037659 
## - Fold07.Rep2: alpha=0.76305, lambda=0.037659 
## + Fold07.Rep2: alpha=0.32620, lambda=0.010929 
## - Fold07.Rep2: alpha=0.32620, lambda=0.010929 
## + Fold07.Rep2: alpha=0.44663, lambda=0.001037 
## - Fold07.Rep2: alpha=0.44663, lambda=0.001037 
## + Fold07.Rep2: alpha=0.08757, lambda=0.012689 
## - Fold07.Rep2: alpha=0.08757, lambda=0.012689 
## + Fold07.Rep2: alpha=0.60040, lambda=0.003594 
## - Fold07.Rep2: alpha=0.60040, lambda=0.003594 
## + Fold07.Rep2: alpha=0.27368, lambda=0.453264 
## - Fold07.Rep2: alpha=0.27368, lambda=0.453264 
## + Fold07.Rep2: alpha=0.36848, lambda=4.961816 
## - Fold07.Rep2: alpha=0.36848, lambda=4.961816 
## + Fold08.Rep2: alpha=0.64457, lambda=0.002748 
## - Fold08.Rep2: alpha=0.64457, lambda=0.002748 
## + Fold08.Rep2: alpha=0.56029, lambda=0.304452 
## - Fold08.Rep2: alpha=0.56029, lambda=0.304452 
## + Fold08.Rep2: alpha=0.34217, lambda=3.683040 
## - Fold08.Rep2: alpha=0.34217, lambda=3.683040 
## + Fold08.Rep2: alpha=0.76305, lambda=0.037659 
## - Fold08.Rep2: alpha=0.76305, lambda=0.037659 
## + Fold08.Rep2: alpha=0.32620, lambda=0.010929 
## - Fold08.Rep2: alpha=0.32620, lambda=0.010929 
## + Fold08.Rep2: alpha=0.44663, lambda=0.001037 
## - Fold08.Rep2: alpha=0.44663, lambda=0.001037 
## + Fold08.Rep2: alpha=0.08757, lambda=0.012689 
## - Fold08.Rep2: alpha=0.08757, lambda=0.012689 
## + Fold08.Rep2: alpha=0.60040, lambda=0.003594 
## - Fold08.Rep2: alpha=0.60040, lambda=0.003594 
## + Fold08.Rep2: alpha=0.27368, lambda=0.453264 
## - Fold08.Rep2: alpha=0.27368, lambda=0.453264 
## + Fold08.Rep2: alpha=0.36848, lambda=4.961816 
## - Fold08.Rep2: alpha=0.36848, lambda=4.961816 
## + Fold09.Rep2: alpha=0.64457, lambda=0.002748 
## - Fold09.Rep2: alpha=0.64457, lambda=0.002748 
## + Fold09.Rep2: alpha=0.56029, lambda=0.304452 
## - Fold09.Rep2: alpha=0.56029, lambda=0.304452 
## + Fold09.Rep2: alpha=0.34217, lambda=3.683040 
## - Fold09.Rep2: alpha=0.34217, lambda=3.683040 
## + Fold09.Rep2: alpha=0.76305, lambda=0.037659 
## - Fold09.Rep2: alpha=0.76305, lambda=0.037659 
## + Fold09.Rep2: alpha=0.32620, lambda=0.010929 
## - Fold09.Rep2: alpha=0.32620, lambda=0.010929 
## + Fold09.Rep2: alpha=0.44663, lambda=0.001037 
## - Fold09.Rep2: alpha=0.44663, lambda=0.001037 
## + Fold09.Rep2: alpha=0.08757, lambda=0.012689 
## - Fold09.Rep2: alpha=0.08757, lambda=0.012689 
## + Fold09.Rep2: alpha=0.60040, lambda=0.003594 
## - Fold09.Rep2: alpha=0.60040, lambda=0.003594 
## + Fold09.Rep2: alpha=0.27368, lambda=0.453264 
## - Fold09.Rep2: alpha=0.27368, lambda=0.453264 
## + Fold09.Rep2: alpha=0.36848, lambda=4.961816 
## - Fold09.Rep2: alpha=0.36848, lambda=4.961816 
## + Fold10.Rep2: alpha=0.64457, lambda=0.002748 
## - Fold10.Rep2: alpha=0.64457, lambda=0.002748 
## + Fold10.Rep2: alpha=0.56029, lambda=0.304452 
## - Fold10.Rep2: alpha=0.56029, lambda=0.304452 
## + Fold10.Rep2: alpha=0.34217, lambda=3.683040 
## - Fold10.Rep2: alpha=0.34217, lambda=3.683040 
## + Fold10.Rep2: alpha=0.76305, lambda=0.037659 
## - Fold10.Rep2: alpha=0.76305, lambda=0.037659 
## + Fold10.Rep2: alpha=0.32620, lambda=0.010929 
## - Fold10.Rep2: alpha=0.32620, lambda=0.010929 
## + Fold10.Rep2: alpha=0.44663, lambda=0.001037 
## - Fold10.Rep2: alpha=0.44663, lambda=0.001037 
## + Fold10.Rep2: alpha=0.08757, lambda=0.012689 
## - Fold10.Rep2: alpha=0.08757, lambda=0.012689 
## + Fold10.Rep2: alpha=0.60040, lambda=0.003594 
## - Fold10.Rep2: alpha=0.60040, lambda=0.003594 
## + Fold10.Rep2: alpha=0.27368, lambda=0.453264 
## - Fold10.Rep2: alpha=0.27368, lambda=0.453264 
## + Fold10.Rep2: alpha=0.36848, lambda=4.961816 
## - Fold10.Rep2: alpha=0.36848, lambda=4.961816 
## + Fold01.Rep3: alpha=0.64457, lambda=0.002748 
## - Fold01.Rep3: alpha=0.64457, lambda=0.002748 
## + Fold01.Rep3: alpha=0.56029, lambda=0.304452 
## - Fold01.Rep3: alpha=0.56029, lambda=0.304452 
## + Fold01.Rep3: alpha=0.34217, lambda=3.683040 
## - Fold01.Rep3: alpha=0.34217, lambda=3.683040 
## + Fold01.Rep3: alpha=0.76305, lambda=0.037659 
## - Fold01.Rep3: alpha=0.76305, lambda=0.037659 
## + Fold01.Rep3: alpha=0.32620, lambda=0.010929 
## - Fold01.Rep3: alpha=0.32620, lambda=0.010929 
## + Fold01.Rep3: alpha=0.44663, lambda=0.001037 
## - Fold01.Rep3: alpha=0.44663, lambda=0.001037 
## + Fold01.Rep3: alpha=0.08757, lambda=0.012689 
## - Fold01.Rep3: alpha=0.08757, lambda=0.012689 
## + Fold01.Rep3: alpha=0.60040, lambda=0.003594 
## - Fold01.Rep3: alpha=0.60040, lambda=0.003594 
## + Fold01.Rep3: alpha=0.27368, lambda=0.453264 
## - Fold01.Rep3: alpha=0.27368, lambda=0.453264 
## + Fold01.Rep3: alpha=0.36848, lambda=4.961816 
## - Fold01.Rep3: alpha=0.36848, lambda=4.961816 
## + Fold02.Rep3: alpha=0.64457, lambda=0.002748 
## - Fold02.Rep3: alpha=0.64457, lambda=0.002748 
## + Fold02.Rep3: alpha=0.56029, lambda=0.304452 
## - Fold02.Rep3: alpha=0.56029, lambda=0.304452 
## + Fold02.Rep3: alpha=0.34217, lambda=3.683040 
## - Fold02.Rep3: alpha=0.34217, lambda=3.683040 
## + Fold02.Rep3: alpha=0.76305, lambda=0.037659 
## - Fold02.Rep3: alpha=0.76305, lambda=0.037659 
## + Fold02.Rep3: alpha=0.32620, lambda=0.010929 
## - Fold02.Rep3: alpha=0.32620, lambda=0.010929 
## + Fold02.Rep3: alpha=0.44663, lambda=0.001037 
## - Fold02.Rep3: alpha=0.44663, lambda=0.001037 
## + Fold02.Rep3: alpha=0.08757, lambda=0.012689 
## - Fold02.Rep3: alpha=0.08757, lambda=0.012689 
## + Fold02.Rep3: alpha=0.60040, lambda=0.003594 
## - Fold02.Rep3: alpha=0.60040, lambda=0.003594 
## + Fold02.Rep3: alpha=0.27368, lambda=0.453264 
## - Fold02.Rep3: alpha=0.27368, lambda=0.453264 
## + Fold02.Rep3: alpha=0.36848, lambda=4.961816 
## - Fold02.Rep3: alpha=0.36848, lambda=4.961816 
## + Fold03.Rep3: alpha=0.64457, lambda=0.002748 
## - Fold03.Rep3: alpha=0.64457, lambda=0.002748 
## + Fold03.Rep3: alpha=0.56029, lambda=0.304452 
## - Fold03.Rep3: alpha=0.56029, lambda=0.304452 
## + Fold03.Rep3: alpha=0.34217, lambda=3.683040 
## - Fold03.Rep3: alpha=0.34217, lambda=3.683040 
## + Fold03.Rep3: alpha=0.76305, lambda=0.037659 
## - Fold03.Rep3: alpha=0.76305, lambda=0.037659 
## + Fold03.Rep3: alpha=0.32620, lambda=0.010929 
## - Fold03.Rep3: alpha=0.32620, lambda=0.010929 
## + Fold03.Rep3: alpha=0.44663, lambda=0.001037 
## - Fold03.Rep3: alpha=0.44663, lambda=0.001037 
## + Fold03.Rep3: alpha=0.08757, lambda=0.012689 
## - Fold03.Rep3: alpha=0.08757, lambda=0.012689 
## + Fold03.Rep3: alpha=0.60040, lambda=0.003594 
## - Fold03.Rep3: alpha=0.60040, lambda=0.003594 
## + Fold03.Rep3: alpha=0.27368, lambda=0.453264 
## - Fold03.Rep3: alpha=0.27368, lambda=0.453264 
## + Fold03.Rep3: alpha=0.36848, lambda=4.961816 
## - Fold03.Rep3: alpha=0.36848, lambda=4.961816 
## + Fold04.Rep3: alpha=0.64457, lambda=0.002748 
## - Fold04.Rep3: alpha=0.64457, lambda=0.002748 
## + Fold04.Rep3: alpha=0.56029, lambda=0.304452 
## - Fold04.Rep3: alpha=0.56029, lambda=0.304452 
## + Fold04.Rep3: alpha=0.34217, lambda=3.683040 
## - Fold04.Rep3: alpha=0.34217, lambda=3.683040 
## + Fold04.Rep3: alpha=0.76305, lambda=0.037659 
## - Fold04.Rep3: alpha=0.76305, lambda=0.037659 
## + Fold04.Rep3: alpha=0.32620, lambda=0.010929 
## - Fold04.Rep3: alpha=0.32620, lambda=0.010929 
## + Fold04.Rep3: alpha=0.44663, lambda=0.001037 
## - Fold04.Rep3: alpha=0.44663, lambda=0.001037 
## + Fold04.Rep3: alpha=0.08757, lambda=0.012689 
## - Fold04.Rep3: alpha=0.08757, lambda=0.012689 
## + Fold04.Rep3: alpha=0.60040, lambda=0.003594 
## - Fold04.Rep3: alpha=0.60040, lambda=0.003594 
## + Fold04.Rep3: alpha=0.27368, lambda=0.453264 
## - Fold04.Rep3: alpha=0.27368, lambda=0.453264 
## + Fold04.Rep3: alpha=0.36848, lambda=4.961816 
## - Fold04.Rep3: alpha=0.36848, lambda=4.961816 
## + Fold05.Rep3: alpha=0.64457, lambda=0.002748 
## - Fold05.Rep3: alpha=0.64457, lambda=0.002748 
## + Fold05.Rep3: alpha=0.56029, lambda=0.304452 
## - Fold05.Rep3: alpha=0.56029, lambda=0.304452 
## + Fold05.Rep3: alpha=0.34217, lambda=3.683040 
## - Fold05.Rep3: alpha=0.34217, lambda=3.683040 
## + Fold05.Rep3: alpha=0.76305, lambda=0.037659 
## - Fold05.Rep3: alpha=0.76305, lambda=0.037659 
## + Fold05.Rep3: alpha=0.32620, lambda=0.010929 
## - Fold05.Rep3: alpha=0.32620, lambda=0.010929 
## + Fold05.Rep3: alpha=0.44663, lambda=0.001037 
## - Fold05.Rep3: alpha=0.44663, lambda=0.001037 
## + Fold05.Rep3: alpha=0.08757, lambda=0.012689 
## - Fold05.Rep3: alpha=0.08757, lambda=0.012689 
## + Fold05.Rep3: alpha=0.60040, lambda=0.003594 
## - Fold05.Rep3: alpha=0.60040, lambda=0.003594 
## + Fold05.Rep3: alpha=0.27368, lambda=0.453264 
## - Fold05.Rep3: alpha=0.27368, lambda=0.453264 
## + Fold05.Rep3: alpha=0.36848, lambda=4.961816 
## - Fold05.Rep3: alpha=0.36848, lambda=4.961816 
## + Fold06.Rep3: alpha=0.64457, lambda=0.002748 
## - Fold06.Rep3: alpha=0.64457, lambda=0.002748 
## + Fold06.Rep3: alpha=0.56029, lambda=0.304452 
## - Fold06.Rep3: alpha=0.56029, lambda=0.304452 
## + Fold06.Rep3: alpha=0.34217, lambda=3.683040 
## - Fold06.Rep3: alpha=0.34217, lambda=3.683040 
## + Fold06.Rep3: alpha=0.76305, lambda=0.037659 
## - Fold06.Rep3: alpha=0.76305, lambda=0.037659 
## + Fold06.Rep3: alpha=0.32620, lambda=0.010929 
## - Fold06.Rep3: alpha=0.32620, lambda=0.010929 
## + Fold06.Rep3: alpha=0.44663, lambda=0.001037 
## - Fold06.Rep3: alpha=0.44663, lambda=0.001037 
## + Fold06.Rep3: alpha=0.08757, lambda=0.012689 
## - Fold06.Rep3: alpha=0.08757, lambda=0.012689 
## + Fold06.Rep3: alpha=0.60040, lambda=0.003594 
## - Fold06.Rep3: alpha=0.60040, lambda=0.003594 
## + Fold06.Rep3: alpha=0.27368, lambda=0.453264 
## - Fold06.Rep3: alpha=0.27368, lambda=0.453264 
## + Fold06.Rep3: alpha=0.36848, lambda=4.961816 
## - Fold06.Rep3: alpha=0.36848, lambda=4.961816 
## + Fold07.Rep3: alpha=0.64457, lambda=0.002748 
## - Fold07.Rep3: alpha=0.64457, lambda=0.002748 
## + Fold07.Rep3: alpha=0.56029, lambda=0.304452 
## - Fold07.Rep3: alpha=0.56029, lambda=0.304452 
## + Fold07.Rep3: alpha=0.34217, lambda=3.683040 
## - Fold07.Rep3: alpha=0.34217, lambda=3.683040 
## + Fold07.Rep3: alpha=0.76305, lambda=0.037659 
## - Fold07.Rep3: alpha=0.76305, lambda=0.037659 
## + Fold07.Rep3: alpha=0.32620, lambda=0.010929 
## - Fold07.Rep3: alpha=0.32620, lambda=0.010929 
## + Fold07.Rep3: alpha=0.44663, lambda=0.001037 
## - Fold07.Rep3: alpha=0.44663, lambda=0.001037 
## + Fold07.Rep3: alpha=0.08757, lambda=0.012689 
## - Fold07.Rep3: alpha=0.08757, lambda=0.012689 
## + Fold07.Rep3: alpha=0.60040, lambda=0.003594 
## - Fold07.Rep3: alpha=0.60040, lambda=0.003594 
## + Fold07.Rep3: alpha=0.27368, lambda=0.453264 
## - Fold07.Rep3: alpha=0.27368, lambda=0.453264 
## + Fold07.Rep3: alpha=0.36848, lambda=4.961816 
## - Fold07.Rep3: alpha=0.36848, lambda=4.961816 
## + Fold08.Rep3: alpha=0.64457, lambda=0.002748 
## - Fold08.Rep3: alpha=0.64457, lambda=0.002748 
## + Fold08.Rep3: alpha=0.56029, lambda=0.304452 
## - Fold08.Rep3: alpha=0.56029, lambda=0.304452 
## + Fold08.Rep3: alpha=0.34217, lambda=3.683040 
## - Fold08.Rep3: alpha=0.34217, lambda=3.683040 
## + Fold08.Rep3: alpha=0.76305, lambda=0.037659 
## - Fold08.Rep3: alpha=0.76305, lambda=0.037659 
## + Fold08.Rep3: alpha=0.32620, lambda=0.010929 
## - Fold08.Rep3: alpha=0.32620, lambda=0.010929 
## + Fold08.Rep3: alpha=0.44663, lambda=0.001037 
## - Fold08.Rep3: alpha=0.44663, lambda=0.001037 
## + Fold08.Rep3: alpha=0.08757, lambda=0.012689 
## - Fold08.Rep3: alpha=0.08757, lambda=0.012689 
## + Fold08.Rep3: alpha=0.60040, lambda=0.003594 
## - Fold08.Rep3: alpha=0.60040, lambda=0.003594 
## + Fold08.Rep3: alpha=0.27368, lambda=0.453264 
## - Fold08.Rep3: alpha=0.27368, lambda=0.453264 
## + Fold08.Rep3: alpha=0.36848, lambda=4.961816 
## - Fold08.Rep3: alpha=0.36848, lambda=4.961816 
## + Fold09.Rep3: alpha=0.64457, lambda=0.002748 
## - Fold09.Rep3: alpha=0.64457, lambda=0.002748 
## + Fold09.Rep3: alpha=0.56029, lambda=0.304452 
## - Fold09.Rep3: alpha=0.56029, lambda=0.304452 
## + Fold09.Rep3: alpha=0.34217, lambda=3.683040 
## - Fold09.Rep3: alpha=0.34217, lambda=3.683040 
## + Fold09.Rep3: alpha=0.76305, lambda=0.037659 
## - Fold09.Rep3: alpha=0.76305, lambda=0.037659 
## + Fold09.Rep3: alpha=0.32620, lambda=0.010929 
## - Fold09.Rep3: alpha=0.32620, lambda=0.010929 
## + Fold09.Rep3: alpha=0.44663, lambda=0.001037 
## - Fold09.Rep3: alpha=0.44663, lambda=0.001037 
## + Fold09.Rep3: alpha=0.08757, lambda=0.012689 
## - Fold09.Rep3: alpha=0.08757, lambda=0.012689 
## + Fold09.Rep3: alpha=0.60040, lambda=0.003594 
## - Fold09.Rep3: alpha=0.60040, lambda=0.003594 
## + Fold09.Rep3: alpha=0.27368, lambda=0.453264 
## - Fold09.Rep3: alpha=0.27368, lambda=0.453264 
## + Fold09.Rep3: alpha=0.36848, lambda=4.961816 
## - Fold09.Rep3: alpha=0.36848, lambda=4.961816 
## + Fold10.Rep3: alpha=0.64457, lambda=0.002748 
## - Fold10.Rep3: alpha=0.64457, lambda=0.002748 
## + Fold10.Rep3: alpha=0.56029, lambda=0.304452 
## - Fold10.Rep3: alpha=0.56029, lambda=0.304452 
## + Fold10.Rep3: alpha=0.34217, lambda=3.683040 
## - Fold10.Rep3: alpha=0.34217, lambda=3.683040 
## + Fold10.Rep3: alpha=0.76305, lambda=0.037659 
## - Fold10.Rep3: alpha=0.76305, lambda=0.037659 
## + Fold10.Rep3: alpha=0.32620, lambda=0.010929 
## - Fold10.Rep3: alpha=0.32620, lambda=0.010929 
## + Fold10.Rep3: alpha=0.44663, lambda=0.001037 
## - Fold10.Rep3: alpha=0.44663, lambda=0.001037 
## + Fold10.Rep3: alpha=0.08757, lambda=0.012689 
## - Fold10.Rep3: alpha=0.08757, lambda=0.012689 
## + Fold10.Rep3: alpha=0.60040, lambda=0.003594 
## - Fold10.Rep3: alpha=0.60040, lambda=0.003594 
## + Fold10.Rep3: alpha=0.27368, lambda=0.453264 
## - Fold10.Rep3: alpha=0.27368, lambda=0.453264 
## + Fold10.Rep3: alpha=0.36848, lambda=4.961816 
## - Fold10.Rep3: alpha=0.36848, lambda=4.961816 
## + Fold01.Rep4: alpha=0.64457, lambda=0.002748 
## - Fold01.Rep4: alpha=0.64457, lambda=0.002748 
## + Fold01.Rep4: alpha=0.56029, lambda=0.304452 
## - Fold01.Rep4: alpha=0.56029, lambda=0.304452 
## + Fold01.Rep4: alpha=0.34217, lambda=3.683040 
## - Fold01.Rep4: alpha=0.34217, lambda=3.683040 
## + Fold01.Rep4: alpha=0.76305, lambda=0.037659 
## - Fold01.Rep4: alpha=0.76305, lambda=0.037659 
## + Fold01.Rep4: alpha=0.32620, lambda=0.010929 
## - Fold01.Rep4: alpha=0.32620, lambda=0.010929 
## + Fold01.Rep4: alpha=0.44663, lambda=0.001037 
## - Fold01.Rep4: alpha=0.44663, lambda=0.001037 
## + Fold01.Rep4: alpha=0.08757, lambda=0.012689 
## - Fold01.Rep4: alpha=0.08757, lambda=0.012689 
## + Fold01.Rep4: alpha=0.60040, lambda=0.003594 
## - Fold01.Rep4: alpha=0.60040, lambda=0.003594 
## + Fold01.Rep4: alpha=0.27368, lambda=0.453264 
## - Fold01.Rep4: alpha=0.27368, lambda=0.453264 
## + Fold01.Rep4: alpha=0.36848, lambda=4.961816 
## - Fold01.Rep4: alpha=0.36848, lambda=4.961816 
## + Fold02.Rep4: alpha=0.64457, lambda=0.002748 
## - Fold02.Rep4: alpha=0.64457, lambda=0.002748 
## + Fold02.Rep4: alpha=0.56029, lambda=0.304452 
## - Fold02.Rep4: alpha=0.56029, lambda=0.304452 
## + Fold02.Rep4: alpha=0.34217, lambda=3.683040 
## - Fold02.Rep4: alpha=0.34217, lambda=3.683040 
## + Fold02.Rep4: alpha=0.76305, lambda=0.037659 
## - Fold02.Rep4: alpha=0.76305, lambda=0.037659 
## + Fold02.Rep4: alpha=0.32620, lambda=0.010929 
## - Fold02.Rep4: alpha=0.32620, lambda=0.010929 
## + Fold02.Rep4: alpha=0.44663, lambda=0.001037 
## - Fold02.Rep4: alpha=0.44663, lambda=0.001037 
## + Fold02.Rep4: alpha=0.08757, lambda=0.012689 
## - Fold02.Rep4: alpha=0.08757, lambda=0.012689 
## + Fold02.Rep4: alpha=0.60040, lambda=0.003594 
## - Fold02.Rep4: alpha=0.60040, lambda=0.003594 
## + Fold02.Rep4: alpha=0.27368, lambda=0.453264 
## - Fold02.Rep4: alpha=0.27368, lambda=0.453264 
## + Fold02.Rep4: alpha=0.36848, lambda=4.961816 
## - Fold02.Rep4: alpha=0.36848, lambda=4.961816 
## + Fold03.Rep4: alpha=0.64457, lambda=0.002748 
## - Fold03.Rep4: alpha=0.64457, lambda=0.002748 
## + Fold03.Rep4: alpha=0.56029, lambda=0.304452 
## - Fold03.Rep4: alpha=0.56029, lambda=0.304452 
## + Fold03.Rep4: alpha=0.34217, lambda=3.683040 
## - Fold03.Rep4: alpha=0.34217, lambda=3.683040 
## + Fold03.Rep4: alpha=0.76305, lambda=0.037659 
## - Fold03.Rep4: alpha=0.76305, lambda=0.037659 
## + Fold03.Rep4: alpha=0.32620, lambda=0.010929 
## - Fold03.Rep4: alpha=0.32620, lambda=0.010929 
## + Fold03.Rep4: alpha=0.44663, lambda=0.001037 
## - Fold03.Rep4: alpha=0.44663, lambda=0.001037 
## + Fold03.Rep4: alpha=0.08757, lambda=0.012689 
## - Fold03.Rep4: alpha=0.08757, lambda=0.012689 
## + Fold03.Rep4: alpha=0.60040, lambda=0.003594 
## - Fold03.Rep4: alpha=0.60040, lambda=0.003594 
## + Fold03.Rep4: alpha=0.27368, lambda=0.453264 
## - Fold03.Rep4: alpha=0.27368, lambda=0.453264 
## + Fold03.Rep4: alpha=0.36848, lambda=4.961816 
## - Fold03.Rep4: alpha=0.36848, lambda=4.961816 
## + Fold04.Rep4: alpha=0.64457, lambda=0.002748 
## - Fold04.Rep4: alpha=0.64457, lambda=0.002748 
## + Fold04.Rep4: alpha=0.56029, lambda=0.304452 
## - Fold04.Rep4: alpha=0.56029, lambda=0.304452 
## + Fold04.Rep4: alpha=0.34217, lambda=3.683040 
## - Fold04.Rep4: alpha=0.34217, lambda=3.683040 
## + Fold04.Rep4: alpha=0.76305, lambda=0.037659 
## - Fold04.Rep4: alpha=0.76305, lambda=0.037659 
## + Fold04.Rep4: alpha=0.32620, lambda=0.010929 
## - Fold04.Rep4: alpha=0.32620, lambda=0.010929 
## + Fold04.Rep4: alpha=0.44663, lambda=0.001037 
## - Fold04.Rep4: alpha=0.44663, lambda=0.001037 
## + Fold04.Rep4: alpha=0.08757, lambda=0.012689 
## - Fold04.Rep4: alpha=0.08757, lambda=0.012689 
## + Fold04.Rep4: alpha=0.60040, lambda=0.003594 
## - Fold04.Rep4: alpha=0.60040, lambda=0.003594 
## + Fold04.Rep4: alpha=0.27368, lambda=0.453264 
## - Fold04.Rep4: alpha=0.27368, lambda=0.453264 
## + Fold04.Rep4: alpha=0.36848, lambda=4.961816 
## - Fold04.Rep4: alpha=0.36848, lambda=4.961816 
## + Fold05.Rep4: alpha=0.64457, lambda=0.002748 
## - Fold05.Rep4: alpha=0.64457, lambda=0.002748 
## + Fold05.Rep4: alpha=0.56029, lambda=0.304452 
## - Fold05.Rep4: alpha=0.56029, lambda=0.304452 
## + Fold05.Rep4: alpha=0.34217, lambda=3.683040 
## - Fold05.Rep4: alpha=0.34217, lambda=3.683040 
## + Fold05.Rep4: alpha=0.76305, lambda=0.037659 
## - Fold05.Rep4: alpha=0.76305, lambda=0.037659 
## + Fold05.Rep4: alpha=0.32620, lambda=0.010929 
## - Fold05.Rep4: alpha=0.32620, lambda=0.010929 
## + Fold05.Rep4: alpha=0.44663, lambda=0.001037 
## - Fold05.Rep4: alpha=0.44663, lambda=0.001037 
## + Fold05.Rep4: alpha=0.08757, lambda=0.012689 
## - Fold05.Rep4: alpha=0.08757, lambda=0.012689 
## + Fold05.Rep4: alpha=0.60040, lambda=0.003594 
## - Fold05.Rep4: alpha=0.60040, lambda=0.003594 
## + Fold05.Rep4: alpha=0.27368, lambda=0.453264 
## - Fold05.Rep4: alpha=0.27368, lambda=0.453264 
## + Fold05.Rep4: alpha=0.36848, lambda=4.961816 
## - Fold05.Rep4: alpha=0.36848, lambda=4.961816 
## + Fold06.Rep4: alpha=0.64457, lambda=0.002748 
## - Fold06.Rep4: alpha=0.64457, lambda=0.002748 
## + Fold06.Rep4: alpha=0.56029, lambda=0.304452 
## - Fold06.Rep4: alpha=0.56029, lambda=0.304452 
## + Fold06.Rep4: alpha=0.34217, lambda=3.683040 
## - Fold06.Rep4: alpha=0.34217, lambda=3.683040 
## + Fold06.Rep4: alpha=0.76305, lambda=0.037659 
## - Fold06.Rep4: alpha=0.76305, lambda=0.037659 
## + Fold06.Rep4: alpha=0.32620, lambda=0.010929 
## - Fold06.Rep4: alpha=0.32620, lambda=0.010929 
## + Fold06.Rep4: alpha=0.44663, lambda=0.001037 
## - Fold06.Rep4: alpha=0.44663, lambda=0.001037 
## + Fold06.Rep4: alpha=0.08757, lambda=0.012689 
## - Fold06.Rep4: alpha=0.08757, lambda=0.012689 
## + Fold06.Rep4: alpha=0.60040, lambda=0.003594 
## - Fold06.Rep4: alpha=0.60040, lambda=0.003594 
## + Fold06.Rep4: alpha=0.27368, lambda=0.453264 
## - Fold06.Rep4: alpha=0.27368, lambda=0.453264 
## + Fold06.Rep4: alpha=0.36848, lambda=4.961816 
## - Fold06.Rep4: alpha=0.36848, lambda=4.961816 
## + Fold07.Rep4: alpha=0.64457, lambda=0.002748 
## - Fold07.Rep4: alpha=0.64457, lambda=0.002748 
## + Fold07.Rep4: alpha=0.56029, lambda=0.304452 
## - Fold07.Rep4: alpha=0.56029, lambda=0.304452 
## + Fold07.Rep4: alpha=0.34217, lambda=3.683040 
## - Fold07.Rep4: alpha=0.34217, lambda=3.683040 
## + Fold07.Rep4: alpha=0.76305, lambda=0.037659 
## - Fold07.Rep4: alpha=0.76305, lambda=0.037659 
## + Fold07.Rep4: alpha=0.32620, lambda=0.010929 
## - Fold07.Rep4: alpha=0.32620, lambda=0.010929 
## + Fold07.Rep4: alpha=0.44663, lambda=0.001037 
## - Fold07.Rep4: alpha=0.44663, lambda=0.001037 
## + Fold07.Rep4: alpha=0.08757, lambda=0.012689 
## - Fold07.Rep4: alpha=0.08757, lambda=0.012689 
## + Fold07.Rep4: alpha=0.60040, lambda=0.003594 
## - Fold07.Rep4: alpha=0.60040, lambda=0.003594 
## + Fold07.Rep4: alpha=0.27368, lambda=0.453264 
## - Fold07.Rep4: alpha=0.27368, lambda=0.453264 
## + Fold07.Rep4: alpha=0.36848, lambda=4.961816 
## - Fold07.Rep4: alpha=0.36848, lambda=4.961816 
## + Fold08.Rep4: alpha=0.64457, lambda=0.002748 
## - Fold08.Rep4: alpha=0.64457, lambda=0.002748 
## + Fold08.Rep4: alpha=0.56029, lambda=0.304452 
## - Fold08.Rep4: alpha=0.56029, lambda=0.304452 
## + Fold08.Rep4: alpha=0.34217, lambda=3.683040 
## - Fold08.Rep4: alpha=0.34217, lambda=3.683040 
## + Fold08.Rep4: alpha=0.76305, lambda=0.037659 
## - Fold08.Rep4: alpha=0.76305, lambda=0.037659 
## + Fold08.Rep4: alpha=0.32620, lambda=0.010929 
## - Fold08.Rep4: alpha=0.32620, lambda=0.010929 
## + Fold08.Rep4: alpha=0.44663, lambda=0.001037 
## - Fold08.Rep4: alpha=0.44663, lambda=0.001037 
## + Fold08.Rep4: alpha=0.08757, lambda=0.012689 
## - Fold08.Rep4: alpha=0.08757, lambda=0.012689 
## + Fold08.Rep4: alpha=0.60040, lambda=0.003594 
## - Fold08.Rep4: alpha=0.60040, lambda=0.003594 
## + Fold08.Rep4: alpha=0.27368, lambda=0.453264 
## - Fold08.Rep4: alpha=0.27368, lambda=0.453264 
## + Fold08.Rep4: alpha=0.36848, lambda=4.961816 
## - Fold08.Rep4: alpha=0.36848, lambda=4.961816 
## + Fold09.Rep4: alpha=0.64457, lambda=0.002748 
## - Fold09.Rep4: alpha=0.64457, lambda=0.002748 
## + Fold09.Rep4: alpha=0.56029, lambda=0.304452 
## - Fold09.Rep4: alpha=0.56029, lambda=0.304452 
## + Fold09.Rep4: alpha=0.34217, lambda=3.683040 
## - Fold09.Rep4: alpha=0.34217, lambda=3.683040 
## + Fold09.Rep4: alpha=0.76305, lambda=0.037659 
## - Fold09.Rep4: alpha=0.76305, lambda=0.037659 
## + Fold09.Rep4: alpha=0.32620, lambda=0.010929 
## - Fold09.Rep4: alpha=0.32620, lambda=0.010929 
## + Fold09.Rep4: alpha=0.44663, lambda=0.001037 
## - Fold09.Rep4: alpha=0.44663, lambda=0.001037 
## + Fold09.Rep4: alpha=0.08757, lambda=0.012689 
## - Fold09.Rep4: alpha=0.08757, lambda=0.012689 
## + Fold09.Rep4: alpha=0.60040, lambda=0.003594 
## - Fold09.Rep4: alpha=0.60040, lambda=0.003594 
## + Fold09.Rep4: alpha=0.27368, lambda=0.453264 
## - Fold09.Rep4: alpha=0.27368, lambda=0.453264 
## + Fold09.Rep4: alpha=0.36848, lambda=4.961816 
## - Fold09.Rep4: alpha=0.36848, lambda=4.961816 
## + Fold10.Rep4: alpha=0.64457, lambda=0.002748 
## - Fold10.Rep4: alpha=0.64457, lambda=0.002748 
## + Fold10.Rep4: alpha=0.56029, lambda=0.304452 
## - Fold10.Rep4: alpha=0.56029, lambda=0.304452 
## + Fold10.Rep4: alpha=0.34217, lambda=3.683040 
## - Fold10.Rep4: alpha=0.34217, lambda=3.683040 
## + Fold10.Rep4: alpha=0.76305, lambda=0.037659 
## - Fold10.Rep4: alpha=0.76305, lambda=0.037659 
## + Fold10.Rep4: alpha=0.32620, lambda=0.010929 
## - Fold10.Rep4: alpha=0.32620, lambda=0.010929 
## + Fold10.Rep4: alpha=0.44663, lambda=0.001037 
## - Fold10.Rep4: alpha=0.44663, lambda=0.001037 
## + Fold10.Rep4: alpha=0.08757, lambda=0.012689 
## - Fold10.Rep4: alpha=0.08757, lambda=0.012689 
## + Fold10.Rep4: alpha=0.60040, lambda=0.003594 
## - Fold10.Rep4: alpha=0.60040, lambda=0.003594 
## + Fold10.Rep4: alpha=0.27368, lambda=0.453264 
## - Fold10.Rep4: alpha=0.27368, lambda=0.453264 
## + Fold10.Rep4: alpha=0.36848, lambda=4.961816 
## - Fold10.Rep4: alpha=0.36848, lambda=4.961816 
## + Fold01.Rep5: alpha=0.64457, lambda=0.002748 
## - Fold01.Rep5: alpha=0.64457, lambda=0.002748 
## + Fold01.Rep5: alpha=0.56029, lambda=0.304452 
## - Fold01.Rep5: alpha=0.56029, lambda=0.304452 
## + Fold01.Rep5: alpha=0.34217, lambda=3.683040 
## - Fold01.Rep5: alpha=0.34217, lambda=3.683040 
## + Fold01.Rep5: alpha=0.76305, lambda=0.037659 
## - Fold01.Rep5: alpha=0.76305, lambda=0.037659 
## + Fold01.Rep5: alpha=0.32620, lambda=0.010929 
## - Fold01.Rep5: alpha=0.32620, lambda=0.010929 
## + Fold01.Rep5: alpha=0.44663, lambda=0.001037 
## - Fold01.Rep5: alpha=0.44663, lambda=0.001037 
## + Fold01.Rep5: alpha=0.08757, lambda=0.012689 
## - Fold01.Rep5: alpha=0.08757, lambda=0.012689 
## + Fold01.Rep5: alpha=0.60040, lambda=0.003594 
## - Fold01.Rep5: alpha=0.60040, lambda=0.003594 
## + Fold01.Rep5: alpha=0.27368, lambda=0.453264 
## - Fold01.Rep5: alpha=0.27368, lambda=0.453264 
## + Fold01.Rep5: alpha=0.36848, lambda=4.961816 
## - Fold01.Rep5: alpha=0.36848, lambda=4.961816 
## + Fold02.Rep5: alpha=0.64457, lambda=0.002748 
## - Fold02.Rep5: alpha=0.64457, lambda=0.002748 
## + Fold02.Rep5: alpha=0.56029, lambda=0.304452 
## - Fold02.Rep5: alpha=0.56029, lambda=0.304452 
## + Fold02.Rep5: alpha=0.34217, lambda=3.683040 
## - Fold02.Rep5: alpha=0.34217, lambda=3.683040 
## + Fold02.Rep5: alpha=0.76305, lambda=0.037659 
## - Fold02.Rep5: alpha=0.76305, lambda=0.037659 
## + Fold02.Rep5: alpha=0.32620, lambda=0.010929 
## - Fold02.Rep5: alpha=0.32620, lambda=0.010929 
## + Fold02.Rep5: alpha=0.44663, lambda=0.001037 
## - Fold02.Rep5: alpha=0.44663, lambda=0.001037 
## + Fold02.Rep5: alpha=0.08757, lambda=0.012689 
## - Fold02.Rep5: alpha=0.08757, lambda=0.012689 
## + Fold02.Rep5: alpha=0.60040, lambda=0.003594 
## - Fold02.Rep5: alpha=0.60040, lambda=0.003594 
## + Fold02.Rep5: alpha=0.27368, lambda=0.453264 
## - Fold02.Rep5: alpha=0.27368, lambda=0.453264 
## + Fold02.Rep5: alpha=0.36848, lambda=4.961816 
## - Fold02.Rep5: alpha=0.36848, lambda=4.961816 
## + Fold03.Rep5: alpha=0.64457, lambda=0.002748 
## - Fold03.Rep5: alpha=0.64457, lambda=0.002748 
## + Fold03.Rep5: alpha=0.56029, lambda=0.304452 
## - Fold03.Rep5: alpha=0.56029, lambda=0.304452 
## + Fold03.Rep5: alpha=0.34217, lambda=3.683040 
## - Fold03.Rep5: alpha=0.34217, lambda=3.683040 
## + Fold03.Rep5: alpha=0.76305, lambda=0.037659 
## - Fold03.Rep5: alpha=0.76305, lambda=0.037659 
## + Fold03.Rep5: alpha=0.32620, lambda=0.010929 
## - Fold03.Rep5: alpha=0.32620, lambda=0.010929 
## + Fold03.Rep5: alpha=0.44663, lambda=0.001037 
## - Fold03.Rep5: alpha=0.44663, lambda=0.001037 
## + Fold03.Rep5: alpha=0.08757, lambda=0.012689 
## - Fold03.Rep5: alpha=0.08757, lambda=0.012689 
## + Fold03.Rep5: alpha=0.60040, lambda=0.003594 
## - Fold03.Rep5: alpha=0.60040, lambda=0.003594 
## + Fold03.Rep5: alpha=0.27368, lambda=0.453264 
## - Fold03.Rep5: alpha=0.27368, lambda=0.453264 
## + Fold03.Rep5: alpha=0.36848, lambda=4.961816 
## - Fold03.Rep5: alpha=0.36848, lambda=4.961816 
## + Fold04.Rep5: alpha=0.64457, lambda=0.002748 
## - Fold04.Rep5: alpha=0.64457, lambda=0.002748 
## + Fold04.Rep5: alpha=0.56029, lambda=0.304452 
## - Fold04.Rep5: alpha=0.56029, lambda=0.304452 
## + Fold04.Rep5: alpha=0.34217, lambda=3.683040 
## - Fold04.Rep5: alpha=0.34217, lambda=3.683040 
## + Fold04.Rep5: alpha=0.76305, lambda=0.037659 
## - Fold04.Rep5: alpha=0.76305, lambda=0.037659 
## + Fold04.Rep5: alpha=0.32620, lambda=0.010929 
## - Fold04.Rep5: alpha=0.32620, lambda=0.010929 
## + Fold04.Rep5: alpha=0.44663, lambda=0.001037 
## - Fold04.Rep5: alpha=0.44663, lambda=0.001037 
## + Fold04.Rep5: alpha=0.08757, lambda=0.012689 
## - Fold04.Rep5: alpha=0.08757, lambda=0.012689 
## + Fold04.Rep5: alpha=0.60040, lambda=0.003594 
## - Fold04.Rep5: alpha=0.60040, lambda=0.003594 
## + Fold04.Rep5: alpha=0.27368, lambda=0.453264 
## - Fold04.Rep5: alpha=0.27368, lambda=0.453264 
## + Fold04.Rep5: alpha=0.36848, lambda=4.961816 
## - Fold04.Rep5: alpha=0.36848, lambda=4.961816 
## + Fold05.Rep5: alpha=0.64457, lambda=0.002748 
## - Fold05.Rep5: alpha=0.64457, lambda=0.002748 
## + Fold05.Rep5: alpha=0.56029, lambda=0.304452 
## - Fold05.Rep5: alpha=0.56029, lambda=0.304452 
## + Fold05.Rep5: alpha=0.34217, lambda=3.683040 
## - Fold05.Rep5: alpha=0.34217, lambda=3.683040 
## + Fold05.Rep5: alpha=0.76305, lambda=0.037659 
## - Fold05.Rep5: alpha=0.76305, lambda=0.037659 
## + Fold05.Rep5: alpha=0.32620, lambda=0.010929 
## - Fold05.Rep5: alpha=0.32620, lambda=0.010929 
## + Fold05.Rep5: alpha=0.44663, lambda=0.001037 
## - Fold05.Rep5: alpha=0.44663, lambda=0.001037 
## + Fold05.Rep5: alpha=0.08757, lambda=0.012689 
## - Fold05.Rep5: alpha=0.08757, lambda=0.012689 
## + Fold05.Rep5: alpha=0.60040, lambda=0.003594 
## - Fold05.Rep5: alpha=0.60040, lambda=0.003594 
## + Fold05.Rep5: alpha=0.27368, lambda=0.453264 
## - Fold05.Rep5: alpha=0.27368, lambda=0.453264 
## + Fold05.Rep5: alpha=0.36848, lambda=4.961816 
## - Fold05.Rep5: alpha=0.36848, lambda=4.961816 
## + Fold06.Rep5: alpha=0.64457, lambda=0.002748 
## - Fold06.Rep5: alpha=0.64457, lambda=0.002748 
## + Fold06.Rep5: alpha=0.56029, lambda=0.304452 
## - Fold06.Rep5: alpha=0.56029, lambda=0.304452 
## + Fold06.Rep5: alpha=0.34217, lambda=3.683040 
## - Fold06.Rep5: alpha=0.34217, lambda=3.683040 
## + Fold06.Rep5: alpha=0.76305, lambda=0.037659 
## - Fold06.Rep5: alpha=0.76305, lambda=0.037659 
## + Fold06.Rep5: alpha=0.32620, lambda=0.010929 
## - Fold06.Rep5: alpha=0.32620, lambda=0.010929 
## + Fold06.Rep5: alpha=0.44663, lambda=0.001037 
## - Fold06.Rep5: alpha=0.44663, lambda=0.001037 
## + Fold06.Rep5: alpha=0.08757, lambda=0.012689 
## - Fold06.Rep5: alpha=0.08757, lambda=0.012689 
## + Fold06.Rep5: alpha=0.60040, lambda=0.003594 
## - Fold06.Rep5: alpha=0.60040, lambda=0.003594 
## + Fold06.Rep5: alpha=0.27368, lambda=0.453264 
## - Fold06.Rep5: alpha=0.27368, lambda=0.453264 
## + Fold06.Rep5: alpha=0.36848, lambda=4.961816 
## - Fold06.Rep5: alpha=0.36848, lambda=4.961816 
## + Fold07.Rep5: alpha=0.64457, lambda=0.002748 
## - Fold07.Rep5: alpha=0.64457, lambda=0.002748 
## + Fold07.Rep5: alpha=0.56029, lambda=0.304452 
## - Fold07.Rep5: alpha=0.56029, lambda=0.304452 
## + Fold07.Rep5: alpha=0.34217, lambda=3.683040 
## - Fold07.Rep5: alpha=0.34217, lambda=3.683040 
## + Fold07.Rep5: alpha=0.76305, lambda=0.037659 
## - Fold07.Rep5: alpha=0.76305, lambda=0.037659 
## + Fold07.Rep5: alpha=0.32620, lambda=0.010929 
## - Fold07.Rep5: alpha=0.32620, lambda=0.010929 
## + Fold07.Rep5: alpha=0.44663, lambda=0.001037 
## - Fold07.Rep5: alpha=0.44663, lambda=0.001037 
## + Fold07.Rep5: alpha=0.08757, lambda=0.012689 
## - Fold07.Rep5: alpha=0.08757, lambda=0.012689 
## + Fold07.Rep5: alpha=0.60040, lambda=0.003594 
## - Fold07.Rep5: alpha=0.60040, lambda=0.003594 
## + Fold07.Rep5: alpha=0.27368, lambda=0.453264 
## - Fold07.Rep5: alpha=0.27368, lambda=0.453264 
## + Fold07.Rep5: alpha=0.36848, lambda=4.961816 
## - Fold07.Rep5: alpha=0.36848, lambda=4.961816 
## + Fold08.Rep5: alpha=0.64457, lambda=0.002748 
## - Fold08.Rep5: alpha=0.64457, lambda=0.002748 
## + Fold08.Rep5: alpha=0.56029, lambda=0.304452 
## - Fold08.Rep5: alpha=0.56029, lambda=0.304452 
## + Fold08.Rep5: alpha=0.34217, lambda=3.683040 
## - Fold08.Rep5: alpha=0.34217, lambda=3.683040 
## + Fold08.Rep5: alpha=0.76305, lambda=0.037659 
## - Fold08.Rep5: alpha=0.76305, lambda=0.037659 
## + Fold08.Rep5: alpha=0.32620, lambda=0.010929 
## - Fold08.Rep5: alpha=0.32620, lambda=0.010929 
## + Fold08.Rep5: alpha=0.44663, lambda=0.001037 
## - Fold08.Rep5: alpha=0.44663, lambda=0.001037 
## + Fold08.Rep5: alpha=0.08757, lambda=0.012689 
## - Fold08.Rep5: alpha=0.08757, lambda=0.012689 
## + Fold08.Rep5: alpha=0.60040, lambda=0.003594 
## - Fold08.Rep5: alpha=0.60040, lambda=0.003594 
## + Fold08.Rep5: alpha=0.27368, lambda=0.453264 
## - Fold08.Rep5: alpha=0.27368, lambda=0.453264 
## + Fold08.Rep5: alpha=0.36848, lambda=4.961816 
## - Fold08.Rep5: alpha=0.36848, lambda=4.961816 
## + Fold09.Rep5: alpha=0.64457, lambda=0.002748 
## - Fold09.Rep5: alpha=0.64457, lambda=0.002748 
## + Fold09.Rep5: alpha=0.56029, lambda=0.304452 
## - Fold09.Rep5: alpha=0.56029, lambda=0.304452 
## + Fold09.Rep5: alpha=0.34217, lambda=3.683040 
## - Fold09.Rep5: alpha=0.34217, lambda=3.683040 
## + Fold09.Rep5: alpha=0.76305, lambda=0.037659 
## - Fold09.Rep5: alpha=0.76305, lambda=0.037659 
## + Fold09.Rep5: alpha=0.32620, lambda=0.010929 
## - Fold09.Rep5: alpha=0.32620, lambda=0.010929 
## + Fold09.Rep5: alpha=0.44663, lambda=0.001037 
## - Fold09.Rep5: alpha=0.44663, lambda=0.001037 
## + Fold09.Rep5: alpha=0.08757, lambda=0.012689 
## - Fold09.Rep5: alpha=0.08757, lambda=0.012689 
## + Fold09.Rep5: alpha=0.60040, lambda=0.003594 
## - Fold09.Rep5: alpha=0.60040, lambda=0.003594 
## + Fold09.Rep5: alpha=0.27368, lambda=0.453264 
## - Fold09.Rep5: alpha=0.27368, lambda=0.453264 
## + Fold09.Rep5: alpha=0.36848, lambda=4.961816 
## - Fold09.Rep5: alpha=0.36848, lambda=4.961816 
## + Fold10.Rep5: alpha=0.64457, lambda=0.002748 
## - Fold10.Rep5: alpha=0.64457, lambda=0.002748 
## + Fold10.Rep5: alpha=0.56029, lambda=0.304452 
## - Fold10.Rep5: alpha=0.56029, lambda=0.304452 
## + Fold10.Rep5: alpha=0.34217, lambda=3.683040 
## - Fold10.Rep5: alpha=0.34217, lambda=3.683040 
## + Fold10.Rep5: alpha=0.76305, lambda=0.037659 
## - Fold10.Rep5: alpha=0.76305, lambda=0.037659 
## + Fold10.Rep5: alpha=0.32620, lambda=0.010929 
## - Fold10.Rep5: alpha=0.32620, lambda=0.010929 
## + Fold10.Rep5: alpha=0.44663, lambda=0.001037 
## - Fold10.Rep5: alpha=0.44663, lambda=0.001037 
## + Fold10.Rep5: alpha=0.08757, lambda=0.012689 
## - Fold10.Rep5: alpha=0.08757, lambda=0.012689 
## + Fold10.Rep5: alpha=0.60040, lambda=0.003594 
## - Fold10.Rep5: alpha=0.60040, lambda=0.003594 
## + Fold10.Rep5: alpha=0.27368, lambda=0.453264 
## - Fold10.Rep5: alpha=0.27368, lambda=0.453264 
## + Fold10.Rep5: alpha=0.36848, lambda=4.961816 
## - Fold10.Rep5: alpha=0.36848, lambda=4.961816

## Warning in nominalTrainWorkflow(x = x, y = y, wts = weights, info = trainInfo, :
## There were missing values in resampled performance measures.

## Aggregating results
## Selecting tuning parameters
## Fitting alpha = 0.326, lambda = 0.0109 on full training set

# Best tuning parameter
elastic_reg$bestTune

##       alpha     lambda
## 3 0.3261968 0.01092878

Metrics for elasticnet

# Make predictions on training set
predictions_train <- predict(elastic_reg, x)
eval_results(y_train, predictions_train, train) 

##        RMSE   Rsquare
## 1 0.6505636 0.3448016

# Make predictions on test set
predictions_test <- predict(elastic_reg, x_test)
eval_results(y_test, predictions_test, test)

##        RMSE   Rsquare
## 1 0.6405588 0.3829841

colnames(df)

##  [1] "fixed.acidity"        "volatile.acidity"     "citric.acid"         
##  [4] "residual.sugar"       "chlorides"            "free.sulfur.dioxide" 
##  [7] "total.sulfur.dioxide" "density"              "pH"                  
## [10] "sulphates"            "alcohol"              "quality"

ggplot(data = df, mapping = aes(x = total.sulfur.dioxide, y = quality)) + 
  geom_point()

ggplot(data = df, mapping = aes(x = residual.sugar, y = quality)) + 
  geom_point()

There is really no relationship. This is probably an exercise in clusterting or machine learning, but linear regression is insufficient.

ggplot(data = df, mapping = aes(x = alcohol, y = quality)) + 
  geom_point() +
  geom_smooth()

## `geom_smooth()` using method = 'gam' and formula 'y ~ s(x, bs = "cs")'